What happens to exponents when multiplying numbers. Properties of degrees: formulations, proofs, examples. Degree with a negative base

The concept of a degree in mathematics is introduced as early as the 7th grade in an algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values ​​and the ability to correctly and quickly count. For faster and better work with mathematics degrees, they came up with the properties of a degree. They help to cut down on big calculations, to convert a huge example into a single number to some extent. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.

degree properties

We will consider 12 properties of a degree, including properties of powers with the same base, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.

1st property.

Many people very often forget about this property, make mistakes, representing a number to the zero degree as zero.

2nd property.

3rd property.

It must be remembered that this property can only be used when multiplying numbers, it does not work with the sum! And we must not forget that this and the following properties apply only to powers with the same base.

4th property.

If the number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in brackets to correctly replace the sign in further calculations.

The property only works when dividing, not when subtracting!

5th property.

6th property.

This property can also be applied in reverse. A unit divided by a number to some degree is that number to a negative power.

7th property.

This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not the properties of the power.

8th property.

9th property.

This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will change depending on the denominator of the degree.

Also, this property is often used in reverse order. The root of any power of a number can be represented as that number to the power of one divided by the power of the root. This property is very useful in cases where the root of the number is not extracted.

10th property.

This property works not only with the square root and the second degree. If the degree of the root and the degree to which this root is raised are the same, then the answer will be a radical expression.

11th property.

You need to be able to see this property in time when solving it in order to save yourself from huge calculations.

12th property.

Each of these properties will meet you more than once in tasks, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of mathematical knowledge.

Application of degrees and their properties

They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as powers often complicate equations and examples related to other sections of mathematics. Exponents help to avoid large and long calculations, it is easier to reduce and calculate the exponents. But to work with large powers, or with powers of large numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time in solving by eliminating the need for long calculations.

The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.

Abbreviated multiplication formulas are another example of the use of powers. They cannot use the properties of degrees, they are decomposed according to special rules, but in each abbreviated multiplication formula there are invariably degrees.

Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and in the future, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations for conversions of units of measurement or calculations of problems, just as in physics, occur using the properties of the degree.

Degrees are also very useful in astronomy, where you can rarely find the use of the properties of a degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.

Degrees are also used in everyday life, when calculating areas, volumes, distances.

With the help of degrees, very large and very small values ​​\u200b\u200bare written in any field of science.

exponential equations and inequalities

Degree properties occupy a special place precisely in exponential equations and inequalities. These tasks are very common, both in the school course and in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, therefore, knowing all the properties, it will not be difficult to solve such an equation or inequality.

Addition and subtraction of powers

Obviously, numbers with powers can be added like other quantities , by adding them one by one with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2 .
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds the same powers of the same variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2 .

It is also obvious that if we take two squares a, or three squares a, or five squares a.

But degrees various variables and various degrees identical variables, must be added by adding them to their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3 .

It is obvious that the square of a, and the cube of a, is neither twice the square of a, but twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6 .

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahend must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Power multiplication

Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same variables.
The expression will take the form: a 5 b 5 y 3 .

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to sum degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n is;

And a m , is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are − negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y-n .y-m = y-n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y).(a + y) = a 2 - y 2 .
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4 .
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8 .

Division of degrees

Numbers with powers can be divided like other numbers by subtracting from the divisor, or by placing them in the form of a fraction.

So a 3 b 2 divided by b 2 is a 3 .

Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing powers with the same base, their exponents are subtracted..

So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.

Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$

It is necessary to master the multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce exponents in $\frac $ Answer: $\frac $.

2. Reduce the exponents in $\frac$. Answer: $\frac $ or 2x.

3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to a common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 / 5a 7 and 5a 5 / 5a 7 or 2a 3 / 5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

degree properties

We remind you that in this lesson we understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.

An exponent with a natural exponent has several important properties that allow you to simplify calculations in exponent examples.

Property #1
Product of powers

When multiplying powers with the same base, the base remains unchanged, and the exponents are added.

a m a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of powers also affects the product of three or more powers.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
• Present as a degree.
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Please note that in the indicated property it was only about multiplying powers with the same bases.. It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5 . This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36 and 3 5 = 243

Property #2
Private degrees

When dividing powers with the same base, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

• Write the quotient as a power
  (2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
• Calculate.

11 3 - 2 4 2 - 1 = 11 4 = 44
Example. Solve the equation. We use the property of partial degrees.
3 8: t = 3 4

Answer: t = 3 4 = 81

Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

Example. Find the value of an expression using degree properties.

2 11 − 5 = 2 6 = 64

Please note that property 2 dealt only with the division of powers with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1 . This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4

Property #3
Exponentiation

When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.

(a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

How to multiply powers

How to multiply powers? Which powers can be multiplied and which cannot? How do you multiply a number by a power?

In algebra, you can find the product of powers in two cases:

1) if the degrees have the same basis;

2) if the degrees have the same indicators.

When multiplying powers with the same base, the base must remain the same, and the exponents must be added:

When multiplying degrees with the same indicators, the total indicator can be taken out of brackets:

Consider how to multiply powers, with specific examples.

The unit in the exponent is not written, but when multiplying the degrees, they take into account:

When multiplying, the number of degrees can be any. It should be remembered that you can not write the multiplication sign before the letter:

In expressions, exponentiation is performed first.

If you need to multiply a number by a power, you must first perform exponentiation, and only then - multiplication:

Multiplying powers with the same base

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In this lesson, we will learn how to multiply powers with the same base. First, we recall the definition of the degree and formulate a theorem on the validity of the equality . Then we give examples of its application to specific numbers and prove it. We will also apply the theorem to solve various problems.

Topic: Degree with a natural indicator and its properties

Lesson: Multiplying powers with the same bases (formula)

1. Basic definitions

Basic definitions:

n- exponent,

— n-th power of a number.

2. Statement of Theorem 1

Theorem 1. For any number a and any natural n and k equality is true:

In other words: if a- any number; n and k natural numbers, then:

Hence rule 1:

3. Explaining tasks

Conclusion: special cases confirmed the correctness of Theorem No. 1. Let us prove it in the general case, that is, for any a and any natural n and k.

4. Proof of Theorem 1

Given a number a- any; numbers n and k- natural. Prove:

The proof is based on the definition of the degree.

5. Solution of examples using Theorem 1

Example 1: Present as a degree.

To solve the following examples, we use Theorem 1.

and)

6. Generalization of Theorem 1

Here is a generalization:

7. Solution of examples using a generalization of Theorem 1

8. Solving various problems using Theorem 1

Example 2: Calculate (you can use the table of basic degrees).

a) (according to the table)

b)

Example 3: Write as a power with base 2.

a)

Example 4: Determine the sign of the number:

, a - negative because the exponent at -13 is odd.

Example 5: Replace ( ) with a power with a base r:

We have , that is .

9. Summing up

1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. et al. Algebra 7. 6th edition. M.: Enlightenment. 2010

1. School Assistant (Source).

1. Express as a degree:

a B C D E)

3. Write as a power with base 2:

4. Determine the sign of the number:

a)

5. Replace ( ) with a power of a number with a base r:

a) r 4 ( ) = r 15 ; b) ( ) r 5 = r 6

Multiplication and division of powers with the same exponents

In this lesson, we will study the multiplication of powers with the same exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising a power to a power. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we will solve a number of typical problems.

Reminder of basic definitions and theorems

Here a- base of degree

— n-th power of a number.

Theorem 1. For any number a and any natural n and k equality is true:

When multiplying powers with the same base, the exponents are added, the base remains unchanged.

Theorem 2. For any number a and any natural n and k, such that n > k equality is true:

When dividing powers with the same base, the exponents are subtracted, and the base remains unchanged.

Theorem 3. For any number a and any natural n and k equality is true:

All the above theorems were about powers with the same grounds, this lesson will consider degrees with the same indicators.

Examples for multiplying powers with the same exponents

Consider the following examples:

Let's write out the expressions for determining the degree.

Conclusion: From the examples, you can see that , but this still needs to be proven. We formulate the theorem and prove it in the general case, that is, for any a and b and any natural n.

Statement and proof of Theorem 4

For any numbers a and b and any natural n equality is true:

Proof Theorem 4 .

By definition of degree:

So we have proven that .

To multiply powers with the same exponent, it is enough to multiply the bases, and leave the exponent unchanged.

Statement and proof of Theorem 5

We formulate a theorem for dividing powers with the same exponents.

For any number a and b() and any natural n equality is true:

Proof Theorem 5 .

Let's write down and by definition of degree:

Statement of theorems in words

So we have proven that .

To divide degrees with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.

Solution of typical problems using Theorem 4

Example 1: Express as a product of powers.

To solve the following examples, we use Theorem 4.

To solve the following example, recall the formulas:

Generalization of Theorem 4

Generalization of Theorem 4:

Solving Examples Using Generalized Theorem 4

Continued solving typical problems

Example 2: Write as a degree of product.

Example 3: Write as a power with an exponent of 2.

Calculation Examples

Example 4: Calculate in the most rational way.

2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF

3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7 .M .: Education. 2006

2. School assistant (Source).

1. Present as a product of powers:

a) ; b) ; in) ; G) ;

2. Write down as the degree of the product:

3. Write in the form of a degree with an indicator of 2:

4. Calculate in the most rational way.

Mathematics lesson on the topic "Multiplication and division of powers"

Sections: Maths

Pedagogical goal:

the student will learn to distinguish between the properties of multiplication and division of powers with a natural exponent; apply these properties in the case of the same bases;

the student will have the opportunity be able to perform transformations of degrees with different bases and be able to perform transformations in combined tasks.

Tasks:

organize the work of students by repeating previously studied material;

ensure the level of reproduction by performing exercises of various types;

organize self-assessment of students through testing.

Activity units of the doctrine: determination of the degree with a natural indicator; degree components; definition of private; associative law of multiplication.

I. Organization of a demonstration of mastering the existing knowledge by students. (step 1)

a) Updating knowledge:

2) Formulate a definition of the degree with a natural indicator.

a n \u003d a a a a ... a (n times)

b k \u003d b b b b a ... b (k times) Justify your answer.

II. Organization of self-assessment of the trainee by the degree of possession of relevant experience. (step 2)

Test for self-examination: (individual work in two versions.)

A1) Express the product 7 7 7 7 x x x as a power:

A2) Express as a product the degree (-3) 3 x 2

A3) Calculate: -2 3 2 + 4 5 3

I select the number of tasks in the test in accordance with the preparation of the class level.

For the test, I give a key for self-testing. Criteria: pass-fail.

III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)

calculate: 2 2 2 3 = ? 3 3 3 2 3 =?

Simplify: a 2 a 20 =? b 30 b 10 b 15 = ?

In the course of solving problems 1) and 2), the students propose a solution, and I, as a teacher, organize a class to find a way to simplify the powers when multiplying with the same bases.

Teacher: come up with a way to simplify powers when multiplying with the same base.

An entry appears on the cluster:

The theme of the lesson is formulated. Multiplication of powers.

Teacher: come up with a rule for dividing degrees with the same bases.

Reasoning: what action checks division? a 5: a 3 = ? that a 2 a 3 = a 5

I return to the scheme - a cluster and supplement the entry - ..when dividing, subtract and add the topic of the lesson. ...and division of degrees.

IV. Communication to students of the limits of knowledge (as a minimum and as a maximum).

Teacher: the task of the minimum for today's lesson is to learn how to apply the properties of multiplication and division of powers with the same bases, and the maximum: to apply multiplication and division together.

Write on the board : a m a n = a m + n ; a m: a n = a m-n

V. Organization of the study of new material. (step 5)

a) According to the textbook: No. 403 (a, c, e) tasks with different wording

No. 404 (a, e, f) independent work, then I organize a mutual check, I give the keys.

b) For what value of m does the equality hold? a 16 a m \u003d a 32; x h x 14 = x 28; x 8 (*) = x 14

Task: come up with similar examples for division.

c) No. 417(a), No. 418 (a) Traps for students: x 3 x n \u003d x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 \u003d a 2.

VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, not teachers, to study this topic) (step 6)

diagnostic work.

Test(place the keys on the back of the test).

Task options: present as a degree the quotient x 15: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 true; find the value of the expression h 0: h 2 with h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .

Summary of the lesson. Reflection. I divide the class into two groups.

Find the arguments of group I: in favor of knowledge of the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers, draw conclusions. In subsequent lessons, you can offer statistical data and name the rubric “It doesn’t fit in my head!”

The average person eats 32 10 2 kg of cucumbers during their lifetime.

The wasp is capable of making a non-stop flight of 3.2 10 2 km.

When glass cracks, the crack propagates at a speed of about 5 10 3 km/h.

A frog eats over 3 tons of mosquitoes in its lifetime. Using the degree, write in kg.

The most prolific is the ocean fish - the moon (Mola mola), which lays up to 300,000,000 eggs with a diameter of about 1.3 mm in one spawning. Write this number using a degree.

VII. Homework.

History reference. What numbers are called Fermat numbers.

P.19. #403, #408, #417

Used Books:

Textbook "Algebra-7", authors Yu.N. Makarychev, N.G. Mindyuk and others.

Didactic material for grade 7, L.V. Kuznetsova, L.I. Zvavich, S.B. Suvorov.

Encyclopedia of Mathematics.

Journal "Quantum".

Properties of degrees, formulations, proofs, examples.

After the degree of the number is determined, it is logical to talk about degree properties. In this article, we will give the basic properties of the degree of a number, while touching on all possible exponents. Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples.

Page navigation.

Properties of degrees with natural indicators

By definition of a power with a natural exponent, the power of a n is the product of n factors, each of which is equal to a . Based on this definition, and using real number multiplication properties, we can obtain and justify the following properties of degree with natural exponent:

the main property of the degree a m ·a n =a m+n , its generalization a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k ;

the property of partial powers with the same bases a m:a n =a m−n ;

product degree property (a b) n =a n b n , its extension (a 1 a 2 a k) n = a 1 n a 2 n a k n ;

quotient property in kind (a:b) n =a n:b n ;

exponentiation (a m) n =a m n , its generalization (((a n 1) n 2) ...) n k =a n 1 ·n 2 ·... n k ;

comparing degree with zero:

• if a>0 , then a n >0 for any natural n ;
• if a=0 , then a n =0 ;
• if a 2 m >0 , if a 2 m−1 n ;
• if m and n are natural numbers such that m>n , then for 0m n , and for a>0 the inequality a m >a n is true.

We immediately note that all the written equalities are identical under the specified conditions, and their right and left parts can be interchanged. For example, the main property of the fraction a m a n = a m + n with simplification of expressions often used in the form a m+n = a m a n .

Now let's look at each of them in detail.

Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.

Let us prove the main property of the degree. By definition of a degree with a natural exponent, the product of powers with the same bases of the form a m a n can be written as the product . Due to the properties of multiplication, the resulting expression can be written as , and this product is the power of a with natural exponent m+n , that is, a m+n . This completes the proof.

Let us give an example that confirms the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, according to the main property of the degree, we can write the equality 2 2 ·2 3 =2 2+3 =2 5 . Let's check its validity, for which we calculate the values ​​of the expressions 2 2 ·2 3 and 2 5 . Performing exponentiation, we have 2 2 2 3 =(2 2) (2 2 2)=4 8=32 and 2 5 =2 2 2 2 2=32 , since we get equal values, then the equality 2 2 2 3 =2 5 is true, and it confirms the main property of the degree.

The main property of a degree based on the properties of multiplication can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 a n 2 a n k =a n 1 +n 2 +…+n k is true.

For example, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 = (2.1) 3+3+4+7 =(2.1) 17 .

You can move on to the next property of degrees with a natural indicator - the property of partial powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n , the equality a m:a n =a m−n is true.

Before giving the proof of this property, let us discuss the meaning of the additional conditions in the statement. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that it is impossible to divide by zero. The condition m>n is introduced so that we do not go beyond natural exponents. Indeed, for m>n, the exponent a m−n is a natural number, otherwise it will be either zero (which happens when m−n) or a negative number (which happens when m m−n a n =a (m−n) + n = a m From the obtained equality a m−n a n = a m and from the relation of multiplication with division it follows that a m−n is a partial power of a m and a n This proves the property of partial powers with the same bases.

Let's take an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the considered property of the degree corresponds to the equality π 5: π 2 = π 5−3 = π 3.

Now consider product degree property: the natural degree n of the product of any two real numbers a and b is equal to the product of the degrees a n and b n , that is, (a b) n =a n b n .

Indeed, by definition of a degree with a natural exponent, we have . The last product, based on the properties of multiplication, can be rewritten as , which is equal to a n b n .

Here's an example: .

This property extends to the degree of the product of three or more factors. That is, the natural degree property n of the product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n .

For clarity, we show this property with an example. For the product of three factors to the power of 7, we have .

The next property is natural property: the quotient of the real numbers a and b , b≠0 to the natural power n is equal to the quotient of the powers a n and b n , that is, (a:b) n =a n:b n .

The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n , and from the equality (a:b) n b n =a n it follows that (a:b) n is a quotient of a n to b n .

Let's write this property using the example of specific numbers: .

Now let's voice exponentiation property: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of a with exponent m·n , that is, (a m) n =a m·n .

For example, (5 2) 3 =5 2 3 =5 6 .

The proof of the power property in a degree is the following chain of equalities: .

The considered property can be extended to degree within degree within degree, and so on. For example, for any natural numbers p, q, r, and s, the equality . For greater clarity, let's give an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10 .

It remains to dwell on the properties of comparing degrees with a natural exponent.

We start by proving the comparison property of zero and power with a natural exponent.

First, let's justify that a n >0 for any a>0 .

The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication allow us to assert that the result of multiplying any number of positive numbers will also be a positive number. And the power of a with natural exponent n is, by definition, the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a the degree of a n is a positive number. By virtue of the proved property 3 5 >0 , (0.00201) 2 >0 and .

It is quite obvious that for any natural n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0 .

Let's move on to negative bases.

Let's start with the case when the exponent is an even number, denote it as 2 m , where m is a natural number. Then . According to the rule of multiplication of negative numbers, each of the products of the form a a is equal to the product of the modules of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive. and degree a 2 m . Here are examples: (−6) 4 >0 , (−2,2) 12 >0 and .

Finally, when the base of a is a negative number and the exponent is an odd number 2 m−1, then . All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. By virtue of this property, (−5) 3 17 n n is the product of the left and right parts of n true inequalities a properties of inequalities, the inequality being proved is of the form a n n . For example, due to this property, the inequalities 3 7 7 and .

It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of the two degrees with natural indicators and the same positive bases, less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree whose indicator is greater is greater. We turn to the proof of this property.

Let us prove that for m>n and 0m n . To do this, we write the difference a m − a n and compare it with zero. The written difference after taking a n out of brackets will take the form a n ·(a m−n −1) . The resulting product is negative as the product of a positive number a n and a negative number a m−n −1 (a n is positive as a natural power of a positive number, and the difference a m−n −1 is negative, since m−n>0 due to the initial condition m>n , whence it follows that for 0m−n it is less than one). Therefore, a m − a n m n , which was to be proved. For example, we give the correct inequality.

It remains to prove the second part of the property. Let us prove that for m>n and a>1, a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree of a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1, the degree of a m−n is greater than one . Therefore, a m − a n >0 and a m >a n , which was to be proved. This property is illustrated by the inequality 3 7 >3 2 .

Properties of degrees with integer exponents

Since positive integers are natural numbers, then all properties of powers with positive integer exponents exactly coincide with the properties of powers with natural exponents listed and proven in the previous paragraph.

We defined a degree with a negative integer exponent, as well as a degree with a zero exponent, so that all properties of degrees with natural exponents expressed by equalities remain valid. Therefore, all these properties are valid both for zero exponents and for negative exponents, while, of course, the bases of the degrees are nonzero.

So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true properties of degrees with integer exponents:

• a m a n \u003d a m + n;
• a m: a n = a m−n ;
• (a b) n = a n b n ;
• (a:b) n =a n:b n ;
• (a m) n = a m n ;
• if n is a positive integer, a and b are positive numbers, and a n n and a−n>b−n ;
• if m and n are integers, and m>n , then for 0m n , and for a>1, the inequality a m >a n is satisfied.

For a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.

It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, we need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p q , (a − p) q =a (−p) q , (a p ) −q =a p (−q) and (a −p) −q =a (−p) (−q) . Let's do it.

For positive p and q, the equality (a p) q =a p·q was proved in the previous subsection. If p=0 , then we have (a 0) q =1 q =1 and a 0 q =a 0 =1 , whence (a 0) q =a 0 q . Similarly, if q=0 , then (a p) 0 =1 and a p 0 =a 0 =1 , whence (a p) 0 =a p 0 . If both p=0 and q=0 , then (a 0) 0 =1 0 =1 and a 0 0 =a 0 =1 , whence (a 0) 0 =a 0 0 .

Let us now prove that (a −p) q =a (−p) q . By definition of a degree with a negative integer exponent , then . By the property of the quotient in the degree, we have . Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the multiplication rules, can be written as a (−p) q .

Similarly .

And .

By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.

In the penultimate of the properties written down, it is worth dwelling on the proof of the inequality a −n >b −n , which is true for any negative integer −n and any positive a and b for which the condition a . We write and transform the difference between the left and right parts of this inequality: . Since by condition a n n , therefore, b n − a n >0 . The product a n ·b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as a quotient of positive numbers b n − a n and a n b n . Hence, whence a −n >b −n , which was to be proved.

The last property of degrees with integer exponents is proved in the same way as the analogous property of degrees with natural exponents.

Properties of powers with rational exponents

We defined the degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, degrees with fractional exponents have the same properties as degrees with integer exponents. Namely:

1. property of the product of powers with the same base for a>0 , and if and , then for a≥0 ;
2. property of partial powers with the same bases for a>0 ;
3. fractional product property for a>0 and b>0 , and if and , then for a≥0 and (or) b≥0 ;
4. quotient property to a fractional power for a>0 and b>0 , and if , then for a≥0 and b>0 ;
5. degree property in degree for a>0 , and if and , then for a≥0 ;
6. the property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is valid, and for p p >b p ;
7. the property of comparing powers with rational exponents and equal bases: for rational numbers p and q, p>q for 0p q, and for a>0, the inequality a p >a q .

The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, on the properties of the arithmetic root of the nth degree, and on the properties of a degree with an integer exponent. Let's give proof.

By definition of the degree with a fractional exponent and , then . The properties of the arithmetic root allow us to write the following equalities. Further, using the property of the degree with an integer exponent, we obtain , whence, by the definition of a degree with a fractional exponent, we have , and the exponent of the degree obtained can be converted as follows: . This completes the proof.

The second property of powers with fractional exponents is proved in exactly the same way:


The rest of the equalities are proved by similar principles:


We turn to the proof of the next property. Let us prove that for any positive a and b , a 0 the inequality a p p is valid, and for p p >b p . We write the rational number p as m/n , where m is an integer and n is a natural number. Conditions p 0 in this case will be equivalent to conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of the roots, we have , and since a and b are positive numbers, then, based on the definition of the degree with a fractional exponent, the resulting inequality can be rewritten as , that is, a p p .

Similarly, when m m >b m , whence , that is, and a p >b p .

It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q . We can always reduce the rational numbers p and q to a common denominator, let us get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from the rule for comparing ordinary fractions with the same denominators. Then, by the property of comparing powers with the same bases and natural exponents, for 0m 1 m 2 , and for a>1, the inequality a m 1 >a m 2 . These inequalities in terms of the properties of the roots can be rewritten, respectively, as and . And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively. From here we draw the final conclusion: for p>q and 0p q , and for a>0, the inequality a p >a q .

Properties of degrees with irrational exponents

From how a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0 , b>0 and irrational numbers p and q the following are true properties of degrees with irrational exponents:

1. a p a q = a p + q ;
2. a p:a q = a p−q ;
3. (a b) p = a p b p ;
4. (a:b) p =a p:b p ;
5. (a p) q = a p q ;
6. for any positive numbers a and b , a 0 the inequality a p p is valid, and for p p >b p ;
7. for irrational numbers p and q , p>q for 0p q , and for a>0 the inequality a p >a q .

From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.

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Earlier we already talked about what a power of a number is. It has certain properties that are useful in solving problems: it is these and all possible exponents that we will analyze in this article. We will also demonstrate with examples how they can be proved and correctly applied in practice.

Let us recall the concept of a degree with a natural exponent that we have already formulated earlier: this is the product of the nth number of factors, each of which is equal to a. We also need to remember how to correctly multiply real numbers. All this will help us to formulate the following properties for a degree with a natural indicator:

Definition 1

1. The main property of the degree: a m a n = a m + n

Can be generalized to: a n 1 · a n 2 · … · a n k = a n 1 + n 2 + … + n k .

2. The quotient property for powers that have the same base: a m: a n = a m − n

3. Product degree property: (a b) n = a n b n

The equality can be extended to: (a 1 a 2 … a k) n = a 1 n a 2 n … a k n

4. Property of a natural degree: (a: b) n = a n: b n

5. We raise the power to the power: (a m) n = a m n ,

Can be generalized to: (((a n 1) n 2) …) n k = a n 1 n 2 … n k

6. Compare the degree with zero:

• if a > 0, then for any natural n, a n will be greater than zero;
• with a equal to 0, a n will also be equal to zero;
• for a< 0 и таком показателе степени, который будет четным числом 2 · m , a 2 · m будет больше нуля;
• for a< 0 и таком показателе степени, который будет нечетным числом 2 · m − 1 , a 2 · m − 1 будет меньше нуля.

7. Equality a n< b n будет справедливо для любого натурального n при условии, что a и b больше нуля и не равны друг другу.

8. The inequality a m > a n will be true provided that m and n are natural numbers, m is greater than n and a is greater than zero and not less than one.

As a result, we got several equalities; if you meet all the conditions indicated above, then they will be identical. For each of the equalities, for example, for the main property, you can swap the right and left parts: a m · a n = a m + n - the same as a m + n = a m · a n . In this form, it is often used when simplifying expressions.

1. Let's start with the basic property of the degree: the equality a m · a n = a m + n will be true for any natural m and n and real a . How to prove this statement?

The basic definition of powers with natural exponents will allow us to convert equality into a product of factors. We will get an entry like this:

This can be shortened to (recall the basic properties of multiplication). As a result, we got the degree of the number a with natural exponent m + n. Thus, a m + n , which means that the main property of the degree is proved.

Let's take a concrete example to prove this.

Example 1

So we have two powers with base 2. Their natural indicators are 2 and 3, respectively. We got the equality: 2 2 2 3 = 2 2 + 3 = 2 5 Let's calculate the values ​​to check the correctness of this equality.

Let's perform the necessary mathematical operations: 2 2 2 3 = (2 2) (2 2 2) = 4 8 = 32 and 2 5 = 2 2 2 2 2 = 32

As a result, we got: 2 2 2 3 = 2 5 . The property has been proven.

Due to the properties of multiplication, we can generalize the property by formulating it in the form of three or more powers, for which the exponents are natural numbers, and the bases are the same. If we denote the number of natural numbers n 1, n 2, etc. by the letter k, we get the correct equality:

a n 1 a n 2 … a n k = a n 1 + n 2 + … + n k .

Example 2

2. Next, we need to prove the following property, which is called the quotient property and is inherent in powers with the same bases: this is the equality a m: a n = a m − n , which is valid for any natural m and n (and m is greater than n)) and any non-zero real a .

To begin with, let us explain what exactly is the meaning of the conditions that are mentioned in the formulation. If we take a equal to zero, then in the end we will get a division by zero, which cannot be done (after all, 0 n = 0). The condition that the number m must be greater than n is necessary so that we can stay within the natural exponents: by subtracting n from m, we get a natural number. If the condition is not met, we will get a negative number or zero, and again we will go beyond the study of degrees with natural indicators.

Now we can move on to the proof. From the previously studied, we recall the basic properties of fractions and formulate the equality as follows:

a m − n a n = a (m − n) + n = a m

From it we can deduce: a m − n a n = a m

Recall the connection between division and multiplication. It follows from it that a m − n is a quotient of powers a m and a n . This is the proof of the second degree property.

Example 3

Substitute specific numbers for clarity in indicators, and denote the base of the degree π: π 5: π 2 = π 5 − 3 = π 3

3. Next, we will analyze the property of the degree of the product: (a · b) n = a n · b n for any real a and b and natural n .

According to the basic definition of a degree with a natural exponent, we can reformulate the equality as follows:

Remembering the properties of multiplication, we write: . It means the same as a n · b n .

Example 4

2 3 - 4 2 5 4 = 2 3 4 - 4 2 5 4

If we have three or more factors, then this property also applies to this case. We introduce the notation k for the number of factors and write:

(a 1 a 2 … a k) n = a 1 n a 2 n … a k n

Example 5

With specific numbers, we get the following correct equality: (2 (- 2 , 3) ​​a) 7 = 2 7 (- 2 , 3) ​​7 a

4. After that, we will try to prove the quotient property: (a: b) n = a n: b n for any real a and b if b is not equal to 0 and n is a natural number.

For the proof, we can use the previous degree property. If (a: b) n b n = ((a: b) b) n = a n , and (a: b) n b n = a n , then it follows that (a: b) n is a quotient of dividing a n by b n .

Example 6

Let's count the example: 3 1 2: - 0 . 5 3 = 3 1 2 3: (- 0 , 5) 3

Example 7

Let's start right away with an example: (5 2) 3 = 5 2 3 = 5 6

And now we formulate a chain of equalities that will prove to us the correctness of the equality:

If we have degrees of degrees in the example, then this property is true for them as well. If we have any natural numbers p, q, r, s, then it will be true:

a p q y s = a p q y s

Example 8

Let's add specifics: (((5 , 2) 3) 2) 5 = (5 , 2) 3 2 5 = (5 , 2) 30

6. Another property of degrees with a natural exponent that we need to prove is the comparison property.

First, let's compare the exponent with zero. Why a n > 0 provided that a is greater than 0?

If we multiply one positive number by another, we will also get a positive number. Knowing this fact, we can say that this does not depend on the number of factors - the result of multiplying any number of positive numbers is a positive number. And what is a degree, if not the result of multiplying numbers? Then for any power a n with a positive base and a natural exponent, this will be true.

Example 9

3 5 > 0 , (0 , 00201) 2 > 0 and 34 9 13 51 > 0

It is also obvious that a power with a base equal to zero is itself zero. To whatever power we raise zero, it will remain so.

Example 10

0 3 = 0 and 0 762 = 0

If the base of the degree is a negative number, then the proof is a little more complicated, since the concept of even / odd exponent becomes important. Let's start with the case when the exponent is even and denote it by 2 · m , where m is a natural number.

Let's remember how to correctly multiply negative numbers: the product a · a is equal to the product of modules, and, therefore, it will be a positive number. Then and the degree a 2 · m are also positive.

Example 11

For example, (− 6) 4 > 0 , (− 2 , 2) 12 > 0 and - 2 9 6 > 0

What if the exponent with a negative base is an odd number? Let's denote it 2 · m − 1 .

Then

All products a · a , according to the properties of multiplication, are positive, and so is their product. But if we multiply it by the only remaining number a , then the final result will be negative.

Then we get: (− 5) 3< 0 , (− 0 , 003) 17 < 0 и - 1 1 102 9 < 0

How to prove it?

a n< b n – неравенство, представляющее собой произведение левых и правых частей nверных неравенств a < b . Вспомним основные свойства неравенств справедливо и a n < b n .

Example 12

For example, the inequalities are true: 3 7< (2 , 2) 7 и 3 5 11 124 > (0 , 75) 124

8. It remains for us to prove the last property: if we have two degrees, the bases of which are the same and positive, and the exponents are natural numbers, then the one of them is greater, the exponent of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree whose indicator is greater is greater.

Let's prove these assertions.

First we need to make sure that a m< a n при условии, что m больше, чем n , и а больше 0 , но меньше 1 .Теперь сравним с нулем разность a m − a n

We take a n out of brackets, after which our difference will take the form a n · (am − n − 1) . Its result will be negative (since the result of multiplying a positive number by a negative one is negative). Indeed, according to the initial conditions, m − n > 0, then a m − n − 1 is negative, and the first factor is positive, like any natural power with a positive base.

It turned out that a m − a n< 0 и a m < a n . Свойство доказано.

It remains to prove the second part of the statement formulated above: a m > a is true for m > n and a > 1 . We indicate the difference and take a n out of brackets: (a m - n - 1) . The power of a n with a greater than one will give a positive result; and the difference itself will also turn out to be positive due to the initial conditions, and for a > 1 the degree of a m − n is greater than one. It turns out that a m − a n > 0 and a m > a n , which is what we needed to prove.

Example 13

Example with specific numbers: 3 7 > 3 2

Basic properties of degrees with integer exponents

For degrees with positive integer exponents, the properties will be similar, because positive integers are natural, which means that all the equalities proved above are also valid for them. They are also suitable for cases where the exponents are negative or equal to zero (provided that the base of the degree itself is non-zero).

Thus, the properties of powers are the same for any bases a and b (provided that these numbers are real and not equal to 0) and any exponents m and n (provided that they are integers). We write them briefly in the form of formulas:

Definition 2

1. a m a n = a m + n

2. a m: a n = a m − n

3. (a b) n = a n b n

4. (a: b) n = a n: b n

5. (am) n = a m n

6. a n< b n и a − n >b − n with positive integer n , positive a and b , a< b

7. a m< a n , при условии целых m и n , m >n and 0< a < 1 , при a >1 a m > a n .

If the base of the degree is equal to zero, then the entries a m and a n make sense only in the case of natural and positive m and n. As a result, we find that the formulations above are also suitable for cases with a degree with a zero base, if all other conditions are met.

The proofs of these properties in this case are simple. We will need to remember what a degree with a natural and integer exponent is, as well as the properties of actions with real numbers.

Let us analyze the property of the degree in the degree and prove that it is true for both positive integers and non-positive integers. We start by proving the equalities (a p) q = a p q , (a − p) q = a (− p) q , (a p) − q = a p (− q) and (a − p) − q = a (−p) (−q)

Conditions: p = 0 or natural number; q - similarly.

If the values ​​of p and q are greater than 0, then we get (a p) q = a p · q . We have already proved a similar equality before. If p = 0 then:

(a 0) q = 1 q = 1 a 0 q = a 0 = 1

Therefore, (a 0) q = a 0 q

For q = 0 everything is exactly the same:

(a p) 0 = 1 a p 0 = a 0 = 1

Result: (a p) 0 = a p 0 .

If both indicators are zero, then (a 0) 0 = 1 0 = 1 and a 0 0 = a 0 = 1, then (a 0) 0 = a 0 0 .

Recall the property of the quotient in the power proved above and write:

1 a p q = 1 q a p q

If 1 p = 1 1 … 1 = 1 and a p q = a p q , then 1 q a p q = 1 a p q

We can transform this notation by virtue of the basic multiplication rules into a (− p) · q .

Also: a p - q = 1 (a p) q = 1 a p q = a - (p q) = a p (- q) .

AND (a - p) - q = 1 a p - q = (a p) q = a p q = a (- p) (- q)

The remaining properties of the degree can be proved in a similar way by transforming the existing inequalities. We will not dwell on this in detail, we will only indicate the difficult points.

Proof of the penultimate property: recall that a − n > b − n is true for any negative integer values ​​of n and any positive a and b, provided that a is less than b .

Then the inequality can be transformed as follows:

1 a n > 1 b n

We write the right and left parts as a difference and perform the necessary transformations:

1 a n - 1 b n = b n - a n a n b n

Recall that in the condition a is less than b , then, according to the definition of a degree with a natural exponent: - a n< b n , в итоге: b n − a n > 0 .

a n · b n ends up being a positive number because its factors are positive. As a result, we have a fraction b n - a n a n · b n , which in the end also gives a positive result. Hence 1 a n > 1 b n whence a − n > b − n , which we had to prove.

The last property of degrees with integer exponents is proved similarly to the property of degrees with natural exponents.

Basic properties of degrees with rational exponents

In previous articles, we discussed what a degree with a rational (fractional) exponent is. Their properties are the same as those of degrees with integer exponents. Let's write:

Definition 3

1. a m 1 n 1 a m 2 n 2 = a m 1 n 1 + m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (product property powers with the same base).

2. a m 1 n 1: b m 2 n 2 = a m 1 n 1 - m 2 n 2 if a > 0 (quotient property).

3. a b m n = a m n b m n for a > 0 and b > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 and (or) b ≥ 0 (product property in fractional degree).

4. a: b m n \u003d a m n: b m n for a > 0 and b > 0, and if m n > 0, then for a ≥ 0 and b > 0 (property of a quotient to a fractional degree).

5. a m 1 n 1 m 2 n 2 \u003d am 1 n 1 m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (degree property in degrees).

6.ap< b p при условии любых положительных a и b , a < b и рациональном p при p >0; if p< 0 - a p >b p (the property of comparing degrees with equal rational exponents).

7.ap< a q при условии рациональных чисел p и q , p >q at 0< a < 1 ; если a >0 – a p > a q

To prove these provisions, we need to remember what a degree with a fractional exponent is, what are the properties of the arithmetic root of the nth degree, and what are the properties of a degree with an integer exponent. Let's take a look at each property.

According to what a degree with a fractional exponent is, we get:

a m 1 n 1 \u003d am 1 n 1 and a m 2 n 2 \u003d am 2 n 2, therefore, a m 1 n 1 a m 2 n 2 \u003d am 1 n 1 a m 2 n 2

The properties of the root will allow us to derive equalities:

a m 1 m 2 n 1 n 2 a m 2 m 1 n 2 n 1 = a m 1 n 2 a m 2 n 1 n 1 n 2

From this we get: a m 1 n 2 a m 2 n 1 n 1 n 2 = a m 1 n 2 + m 2 n 1 n 1 n 2

Let's transform:

a m 1 n 2 a m 2 n 1 n 1 n 2 = a m 1 n 2 + m 2 n 1 n 1 n 2

The exponent can be written as:

m 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 2 n 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 1 + m 2 n 2

This is the proof. The second property is proved in exactly the same way. Let's write down the chain of equalities:

a m 1 n 1: a m 2 n 2 = a m 1 n 1: a m 2 n 2 = a m 1 n 2: a m 2 n 1 n 1 n 2 = = a m 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 2 n 1 n 2 - m 2 n 1 n 1 n 2 = a m 1 n 1 - m 2 n 2

Proofs of the remaining equalities:

a b m n = (a b) m n = a m b m n = a m n b m n = a m n b m n ; (a: b) m n = (a: b) m n = a m: b m n = = a m n: b m n = a m n: b m n ; a m 1 n 1 m 2 n 2 = a m 1 n 1 m 2 n 2 = a m 1 n 1 m 2 n 2 = = a m 1 m 2 n 1 n 2 = a m 1 m 2 n 1 n 2 = = a m 1 m 2 n 2 n 1 = a m 1 m 2 n 2 n 1 = a m 1 n 1 m 2 n 2

Next property: let's prove that for any values ​​of a and b greater than 0 , if a is less than b , a p will be executed< b p , а для p больше 0 - a p >bp

Let's represent a rational number p as m n . In this case, m is an integer, n is a natural number. Then the conditions p< 0 и p >0 will be extended to m< 0 и m >0 . For m > 0 and a< b имеем (согласно свойству степени с целым положительным показателем), что должно выполняться неравенство a m < b m .

We use the property of roots and derive: a m n< b m n

Taking into account the positiveness of the values ​​a and b , we rewrite the inequality as a m n< b m n . Оно эквивалентно a p < b p .

In the same way, for m< 0 имеем a a m >b m , we get a m n > b m n so a m n > b m n and a p > b p .

It remains for us to prove the last property. Let us prove that for rational numbers p and q , p > q at 0< a < 1 a p < a q , а при a >0 would be true a p > a q .

Rational numbers p and q can be reduced to a common denominator and get fractions m 1 n and m 2 n

Here m 1 and m 2 are integers, and n is a natural number. If p > q, then m 1 > m 2 (taking into account the rule for comparing fractions). Then at 0< a < 1 будет верно a m 1 < a m 2 , а при a >1 – inequality a 1 m > a 2 m .

They can be rewritten in the following form:

a m 1 n< a m 2 n a m 1 n >a m 2 n

Then you can make transformations and get as a result:

a m 1 n< a m 2 n a m 1 n >a m 2 n

To summarize: for p > q and 0< a < 1 верно a p < a q , а при a >0 – a p > a q .

Basic properties of degrees with irrational exponents

All the properties described above that a degree with rational exponents possesses can be extended to such a degree. This follows from its very definition, which we gave in one of the previous articles. Let us briefly formulate these properties (conditions: a > 0 , b > 0 , indicators p and q are irrational numbers):

Definition 4

1. a p a q = a p + q

2. a p: a q = a p − q

3. (a b) p = a p b p

4. (a: b) p = a p: b p

5. (a p) q = a p q

6.ap< b p верно при любых положительных a и b , если a < b и p – иррациональное число больше 0 ; если p меньше 0 , то a p >bp

7.ap< a q верно, если p и q – иррациональные числа, p < q , 0 < a < 1 ; если a >0 , then a p > a q .

Thus, all powers whose exponents p and q are real numbers, provided that a > 0, have the same properties.

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were swapped, the rule could apply.

But how to do that? It turns out that it is very easy: the even degree of the denominator helps us here.

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

whole we name the natural numbers, their opposites (that is, taken with the sign "") and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, we ask ourselves: why is this so?

Consider some power with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. What number must be multiplied by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

Let's go further. In addition to natural numbers and numbers, integers include negative numbers. To understand what a negative degree is, let's do the same as last time: we multiply some normal number by the same in a negative degree:

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

A number to a negative power is the inverse of the same number to a positive power. But at the same time base cannot be null:(because it is impossible to divide).

Let's summarize:

I. Expression is not defined in case. If, then.

II. Any number to the zero power is equal to one: .

III. A number that is not equal to zero to a negative power is the inverse of the same number to a positive power: .

Tasks for independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in the exam!

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is "fractional degree" Let's consider a fraction:

Let's raise both sides of the equation to a power:

Now remember the rule "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal.

That is, the root of the th degree is the inverse operation of exponentiation: .

It turns out that. Obviously, this special case can be extended: .

Now add the numerator: what is it? The answer is easy to get with the power-to-power rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here a problem arises.

The number can be represented as other, reduced fractions, for example, or.

And it turns out that it exists, but does not exist, and these are just two different records of the same number.

Or another example: once, then you can write it down. But as soon as we write the indicator in a different way, we again get trouble: (that is, we got a completely different result!).

To avoid such paradoxes, consider only positive base exponent with fractional exponent.

So if:

• - natural number;
• is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

1. Do not forget about the usual properties of degrees:

2. . Here we recall that we forgot to learn the table of degrees:

after all - this or. The solution is found automatically: .

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

All the rules and properties of degrees here are exactly the same as for degrees with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero power- this is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number;

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number.

But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a degree to a degree:

Now look at the score. Does he remind you of anything? We recall the formula for abbreviated multiplication of the difference of squares:

In this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal or both ordinary. We get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Definition of degree

The degree is an expression of the form: , where:

• — base of degree;
• - exponent.

Degree with natural exponent (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

• - natural number;
• is an integer;

Examples:

Degree properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

By definition:

So, on the right side of this expression, the following product is obtained:

But by definition, this is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule necessarily must have the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

Another important note: this rule - only for products of powers!

Under no circumstances should I write that.

Just as with the previous property, let's turn to the definition of the degree:

Let's rearrange it like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the -th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be index degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

Indeed, we can multiply any number by each other, whether they are positive, negative, or even. Let's think about what signs (" " or "") will have degrees of positive and negative numbers?

For example, will the number be positive or negative? BUT? ?

With the first, everything is clear: no matter how many positive numbers we multiply with each other, the result will be positive.

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a minus gives a plus.” That is, or. But if we multiply by (), we get -.

And so on ad infinitum: with each subsequent multiplication, the sign will change. You can formulate these simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any power is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent, and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it doesn’t matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them into each other, divide them into pairs and get:

Before analyzing the last rule, let's solve a few examples.

Calculate the values ​​of expressions:

Solutions :

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares!

We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were reversed, rule 3 could be applied. But how to do this? It turns out that it is very easy: the even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it looks like this:

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced by by changing only one objectionable minus to us!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average level, we will analyze the degree with an irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with an integer negative indicator - it is as if a certain “reverse process” has occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians have created to extend the concept of a degree to the entire space of numbers.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number. But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

1. Remember the difference of squares formula. Answer: .
2. We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
3. Nothing special, we apply the usual properties of degrees:

SECTION SUMMARY AND BASIC FORMULA

Degree is called an expression of the form: , where:

Degree with integer exponent

degree, the exponent of which is a natural number (i.e. integer and positive).

Degree with rational exponent

degree, the indicator of which is negative and fractional numbers.

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree properties

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any power is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

One of the main characteristics in algebra, and indeed in all mathematics, is a degree. Of course, in the 21st century, all calculations can be carried out on an online calculator, but it is better to learn how to do it yourself for the development of brains.

In this article, we will consider the most important issues regarding this definition. Namely, we will understand what it is in general and what are its main functions, what properties exist in mathematics.

Let's look at examples of what the calculation looks like, what are the basic formulas. We will analyze the main types of quantities and how they differ from other functions.

We will understand how to solve various problems using this value. We will show with examples how to raise to a zero degree, irrational, negative, etc.

Online exponentiation calculator

What is the degree of a number

What is meant by the expression "raise a number to a power"?

The degree n of a number a is the product of factors of magnitude a n times in a row.

Mathematically it looks like this:

a n = a * a * a * …a n .

For example:

• 2 3 = 2 in the third step. = 2 * 2 * 2 = 8;
• 4 2 = 4 in step. two = 4 * 4 = 16;
• 5 4 = 5 in step. four = 5 * 5 * 5 * 5 = 625;
• 10 5 \u003d 10 in 5 step. = 10 * 10 * 10 * 10 * 10 = 100000;
• 10 4 \u003d 10 in 4 step. = 10 * 10 * 10 * 10 = 10000.

Below is a table of squares and cubes from 1 to 10.

Table of degrees from 1 to 10

Below are the results of raising natural numbers to positive powers - "from 1 to 100".

Ch-lo	2nd grade	3rd grade
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	279
10	100	1000

Degree properties

What is characteristic of such a mathematical function? Let's look at the basic properties.

Scientists have established the following signs characteristic of all degrees:

• a n * a m = (a) (n+m) ;
• a n: a m = (a) (n-m) ;
• (a b) m =(a) (b*m) .

Let's check with examples:

2 3 * 2 2 = 8 * 4 = 32. On the other hand 2 5 = 2 * 2 * 2 * 2 * 2 = 32.

Similarly: 2 3: 2 2 = 8 / 4 = 2. Otherwise 2 3-2 = 2 1 =2.

(2 3) 2 = 8 2 = 64. What if it's different? 2 6 = 2 * 2 * 2 * 2 * 2 * 2 = 32 * 2 = 64.

As you can see, the rules work.

But how to be with addition and subtraction? Everything is simple. First exponentiation is performed, and only then addition and subtraction.

Let's look at examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 - 3 2 = 25 - 9 = 16

But in this case, you must first calculate the addition, since there are actions in brackets: (5 + 3) 3 = 8 3 = 512.

How to produce calculations in more complex cases? The order is the same:

• if there are brackets, you need to start with them;
• then exponentiation;
• then perform operations of multiplication, division;
• after addition, subtraction.

There are specific properties that are not characteristic of all degrees:

1. The root of the nth degree from the number a to the degree m will be written as: a m / n .
2. When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
3. When raising the product of different numbers to a power, the expression will correspond to the product of these numbers to a given power. That is: (a * b) n = a n * b n .
4. When raising a number to a negative power, you need to divide 1 by a number in the same step, but with a “+” sign.
5. If the denominator of a fraction is in a negative power, then this expression will be equal to the product of the numerator and the denominator in a positive power.
6. Any number to the power of 0 = 1, and to the step. 1 = to himself.

These rules are important in individual cases, we will consider them in more detail below.

Degree with a negative exponent

What to do with a negative degree, that is, when the indicator is negative?

Based on properties 4 and 5(see point above) it turns out:

A (- n) \u003d 1 / A n, 5 (-2) \u003d 1/5 2 \u003d 1/25.

And vice versa:

1 / A (- n) \u003d A n, 1 / 2 (-3) \u003d 2 3 \u003d 8.

What if it's a fraction?

(A / B) (- n) = (B / A) n , (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.

Degree with a natural indicator

It is understood as a degree with exponents equal to integers.

Things to remember:

A 0 = 1, 1 0 = 1; 2 0 = 1; 3.15 0 = 1; (-4) 0 = 1…etc.

A 1 = A, 1 1 = 1; 2 1 = 2; 3 1 = 3…etc.

Also, if (-a) 2 n +2 , n=0, 1, 2…then the result will be with a “+” sign. If a negative number is raised to an odd power, then vice versa.

General properties, and all the specific features described above, are also characteristic of them.

Fractional degree

This view can be written as a scheme: A m / n. It is read as: the root of the nth degree of the number A to the power of m.

With a fractional indicator, you can do anything: reduce, decompose into parts, raise to another degree, etc.

Degree with irrational exponent

Let α be an irrational number and А ˃ 0.

To understand the essence of the degree with such an indicator, Let's look at different possible cases:

• A \u003d 1. The result will be equal to 1. Since there is an axiom - 1 is equal to one in all powers;

А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 are rational numbers;

• 0˂А˂1.

In this case, vice versa: А r 2 ˂ А α ˂ А r 1 under the same conditions as in the second paragraph.

For example, the exponent is the number π. It is rational.

r 1 - in this case it is equal to 3;

r 2 - will be equal to 4.

Then, for A = 1, 1 π = 1.

A = 2, then 2 3 ˂ 2 π ˂ 2 4 , 8 ˂ 2 π ˂ 16.

A = 1/2, then (½) 4 ˂ (½) π ˂ (½) 3 , 1/16 ˂ (½) π ˂ 1/8.

Such degrees are characterized by all the mathematical operations and specific properties described above.

Conclusion

Let's summarize - what are these values ​​for, what are the advantages of such functions? Of course, first of all, they simplify the lives of mathematicians and programmers when solving examples, since they allow minimizing calculations, reducing algorithms, systematizing data, and much more.

Where else can this knowledge be useful? In any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.