Properties of degrees: statements, proofs, examples. Properties of degrees, formulations, proofs, examples Properties of degrees with natural exponent rules

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

Let us recall the concept of a degree with a natural exponent, already formulated by us earlier: this is the product of an n-number of factors, each of which is equal to a. We also need to remember how to multiply real numbers correctly. 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 \u003d a m + n

Can be generalized to: a n 1 · a n 2 ·… · a n k \u003d a n 1 + n 2 +… + n k.

2. Property of the quotient for degrees with the same bases: a m: a n \u003d a m - n

3. The property of the degree of the product: (a b) n \u003d a n b n

Equality can be extended to: (a 1 a 2… a k) n \u003d a 1 n a 2 n… a k n

4. Property of the quotient in natural degree: (a: b) n \u003d a n: b n

5. Raise the power to the power: (a m) n \u003d a m · n,

Can be generalized to: (((a n 1) n 2)…) n k \u003d a n 1 n 2… n k

6. Compare the degree with zero:

• if a\u003e 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;
• at a< 0 и таком показателе степени, который будет четным числом 2 · m , a 2 · m будет больше нуля;
• at 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\u003e 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 all the conditions indicated above are met, they will be identical. For each of the equalities, for example, for the main property, you can swap the right and left sides: a m · a n \u003d a m + n - the same as a m + n \u003d a m · a n. As such, it is often used to simplify expressions.

1. Let's start with the main property of the degree: the equality a m · a n \u003d a m + n will be true for any natural m and n and a real a. How can you prove this statement?

The basic definition of degrees with natural exponents will allow us to convert equality to a product of factors. We will get a record like this:

This can be shortened to (remember the basic properties of multiplication). As a result, we got the power 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 look at a specific example that confirms this.

Example 1

So we have two degrees with base 2. Their natural indicators are 2 and 3, respectively. We got an equality: 2 2 · 2 3 \u003d 2 2 + 3 \u003d 2 5 Let's calculate the values \u200b\u200bto check if this equality is correct.

Let's perform the necessary mathematical operations: 2 2 2 3 \u003d (2 2) (2 2 2) \u003d 4 8 \u003d 32 and 2 5 \u003d 2 2 2 2 2 \u003d 32

As a result, we got: 2 2 2 3 \u003d 2 5. The property is proven.

Due to the properties of multiplication, we can generalize the property by formulating it in the form of three or more degrees, whose 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 \u003d a n 1 + n 2 +… + n k.

Example 2

2. Next, we need to prove the following property, which is called the property of the quotient and is inherent in degrees with the same bases: this is the equality am: an \u003d am - n, which is valid for any natural numbers m and n (where m is greater than n)) and any nonzero real a ...

To begin with, let us explain exactly what is the meaning of the conditions that are mentioned in the wording. If we take a equal to zero, then in the end we get division by zero, which cannot be done (after all, 0 n \u003d 0). The condition that the number m must necessarily be greater than n is necessary so that we can stay within the natural exponents: subtracting n from m, we get natural number... If the condition is not met, we will end up with a negative number or zero, and again we will go beyond studying degrees with natural indicators.

We can now move on to the proof. From what we studied earlier, we recall the basic properties of fractions and formulate the equality as follows:

a m - n a n \u003d a (m - n) + n \u003d a m

From it you can deduce: a m - n a n \u003d a m

Let's remember the connection between division and multiplication. It follows from it that a m - n is a quotient of degrees a m and a n. This is the proof of the second property of the degree.

Example 3

Substitute specific numbers for clarity in the indicators, and denote the base of the degree by π: π 5: π 2 \u003d π 5 - 3 \u003d π 3

3. Next, we will analyze the property of the degree of the product: (a b) n \u003d 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: ... This means the same as a n · b n.

Example 4

2 3 - 4 2 5 4 \u003d 2 3 4 - 4 2 5 4

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

(a 1 a 2… a k) n \u003d a 1 n a 2 n… a k n

Example 5

With specific numbers, we obtain the following true equality: (2 (- 2, 3) a) 7 \u003d 2 7 (- 2, 3) 7 a

4. After that, we will try to prove the property of the quotient: (a: b) n \u003d 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, you can use the previous property of the degree. If (a: b) n bn \u003d ((a: b) b) n \u003d an, and (a: b) n bn \u003d an, then this implies that (a: b) n is the quotient of dividing an by bn.

Example 6

Let's calculate an example: 3 1 2: - 0. 5 3 \u003d 3 1 2 3: (- 0, 5) 3

Example 7

Let's start right away with an example: (5 2) 3 \u003d 5 2 3 \u003d 5 6

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

If we have degrees of degrees in our 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 \u003d a p q y s

Example 8

Add specifics: (((5, 2) 3) 2) 5 \u003d (5, 2) 3 2 5 \u003d (5, 2) 30

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

First, let's compare the degree to zero. Why a n\u003e 0, provided that a is greater than 0?

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

Example 9

3 5\u003e 0, (0, 00201) 2\u003e 0 and 34 9 13 51\u003e 0

It is also obvious that a degree with base equal to zero is itself zero. No matter what degree we raise zero, it will remain.

Example 10

0 3 \u003d 0 and 0 762 \u003d 0

If the base of the exponent is a negative number, then the proof is a little more difficult, since the notion of even / odd exponent becomes important. First, take the case when the exponent is even and denote it 2 · m, where m is a natural number.

Let's remember how to multiply negative numbers correctly: 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\u003e 0, (- 2, 2) 12\u003e 0 and - 2 9 6\u003e 0

What if the exponent with a negative base is an odd number? We denote it 2 m - 1.

Then

All products a · a, according to the properties of multiplication, are positive, their product is also. 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 greater is the degree whose indicator is greater.

Let us prove these statements.

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

Let's take a n out of the brackets, after which our difference will take the form a n · (a m - n - 1). Its result will be negative (since the result of multiplying a positive number by a negative number is negative). After all, according to initial conditions, m - n\u003e 0, then a m - n - 1 is negative, and the first factor is positive, like any natural degree with a positive base.

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

It remains to give a proof of the second part of the statement formulated above: a m\u003e a is valid for m\u003e n and a\u003e 1. Let us indicate the difference and put a n outside the brackets: (a m - n - 1). The degree of a n for a greater than one will give a positive result; and the difference itself also turns out to be positive due to the initial conditions, and for a\u003e 1 the degree of a m - n is greater than one. It turns out that a m - a n\u003e 0 and a m\u003e a n, which is what we needed to prove.

Example 13

Example with specific numbers: 3 7\u003e 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 true 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 nonzero).

Thus, the properties of the degrees 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). Let's write them briefly in the form of formulas:

Definition 2

1.a m a n \u003d a m + n

2.a m: a n \u003d a m - n

3. (a b) n \u003d a n b n

4. (a: b) n \u003d a n: b n

5. (a m) n \u003d a m n

6.a n< b n и a − n > b - n assuming a 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\u003e a n.

If the base of the degree is equal to zero, then the notations 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 base zero, if all other conditions are met.

The proofs of these properties in this case are not complicated. We need to remember what a degree with natural and integer exponents is, as well as the properties of actions with real numbers.

Let us analyze the property of degree to degree and prove that it is true for both positive and non-positive integers. We start by proving the equalities (ap) q \u003d ap q, (a - p) q \u003d a (- p) q, (ap) - q \u003d ap (- q), and (a - p) - q \u003d a (- p) (- q)

Conditions: p \u003d 0 or natural number; q - similarly.

If the values \u200b\u200bof p and q are greater than 0, then we get (a p) q \u003d a p q. We have already proved a similar equality earlier. If p \u003d 0, then:

(a 0) q \u003d 1 q \u003d 1 a 0 q \u003d a 0 \u003d 1

Therefore, (a 0) q \u003d a 0 q

For q \u003d 0, everything is exactly the same:

(a p) 0 \u003d 1 a p 0 \u003d a 0 \u003d 1

Result: (a p) 0 \u003d a p · 0.

If both exponents are zero, then (a 0) 0 \u003d 1 0 \u003d 1 and a 0 · 0 \u003d a 0 \u003d 1, which means (a 0) 0 \u003d a 0 · 0.

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

1 a p q \u003d 1 q a p q

If 1 p \u003d 1 1 ... 1 \u003d 1 and a p q \u003d a p q, then 1 q a p q \u003d 1 a p q

We can transform this notation into a (- p) q due to the basic rules of multiplication.

Likewise: a p - q \u003d 1 (a p) q \u003d 1 a p q \u003d a - (p q) \u003d a p (- q).

And (a - p) - q \u003d 1 a p - q \u003d (a p) q \u003d a p q \u003d a (- p) (- q)

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

Proof of the penultimate property: recall that a - n\u003e b - n is true for any negative integer values \u200b\u200bof 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\u003e 1 b n

Let's write the right and left parts as a difference and perform the necessary transformations:

1 a n - 1 b n \u003d 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 with 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\u003e 1 b n whence a - n\u003e b - n, which we needed 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 indicators

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.am 1 n 1 am 2 n 2 \u003d am 1 n 1 + m 2 n 2 for a\u003e 0, and if m 1 n 1\u003e 0 and m 2 n 2\u003e 0, then for a ≥ 0 (property of the product degrees with the same bases).

2.a m 1 n 1: b m 2 n 2 \u003d a m 1 n 1 - m 2 n 2, if a\u003e 0 (property of the quotient).

3.a bmn \u003d amn bmn for a\u003e 0 and b\u003e 0, and if m 1 n 1\u003e 0 and m 2 n 2\u003e 0, then for a ≥ 0 and (or) b ≥ 0 (property of the product in fractional degree).

4.a: b m n \u003d a m n: b m n for a\u003e 0 and b\u003e 0, and if m n\u003e 0, then for a ≥ 0 and b\u003e 0 (property of the quotient in fractional power).

5.am 1 n 1 m 2 n 2 \u003d am 1 n 1 m 2 n 2 for a\u003e 0, and if m 1 n 1\u003e 0 and m 2 n 2\u003e 0, then for a ≥ 0 (property of degree in degree).

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

7.a p< a q при условии рациональных чисел p и q , p > q at 0< a < 1 ; если a > 0 - a p\u003e a q

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

According to what a fractional exponent is, we get:

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

The root properties allow us to deduce equalities:

a m 1 m 2 n 1 n 2 a m 2 m 1 n 2 n 1 \u003d 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 \u003d 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 \u003d 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 \u003d m 1 n 2 n 1 n 2 + m 2 n 1 n 1 n 2 \u003d 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:

am 1 n 1: am 2 n 2 \u003d am 1 n 1: am 2 n 2 \u003d am 1 n 2: am 2 n 1 n 1 n 2 \u003d \u003d am 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 2 n 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 1 - m 2 n 2

Proofs of the remaining equalities:

a b m n \u003d (a b) m n \u003d a m b m n \u003d a m n b m n \u003d a m n b m n; (a: b) m n \u003d (a: b) m n \u003d a m: b m n \u003d \u003d a m n: b m n \u003d a m n: b m n; am 1 n 1 m 2 n 2 \u003d am 1 n 1 m 2 n 2 \u003d am 1 n 1 m 2 n 2 \u003d \u003d am 1 m 2 n 1 n 2 \u003d am 1 m 2 n 1 n 2 \u003d \u003d am 1 M 2 n 2 n 1 \u003d am 1 m 2 n 2 n 1 \u003d am 1 n 1 m 2 n 2

The next property: we prove that for any values \u200b\u200bof a and b greater than 0, if a is less than b, then a p< b p , а для p больше 0 - a p > b p

We represent the rational number p as m n. In this case, m is an integer, n is natural. Then the conditions p< 0 и p > 0 will extend to m< 0 и m > 0. For m\u003e 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 output: a m n< b m n

Given the positive values \u200b\u200bof 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\u003e b m n which means that a m n\u003e b m n and a p\u003e b p.

It remains for us to give a proof of the last property. Let us prove that for rational numbers p and q, p\u003e q for 0< a < 1 a p < a q , а при a > 0 will be true a p\u003e 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 natural. If p\u003e q, then m 1\u003e 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\u003e a 2 m.

They can be rewritten as follows:

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\u003e q and 0< a < 1 верно a p < a q , а при a > 0 - a p\u003e a q.

Basic properties of degrees with irrational exponents

All the properties described above that a degree with rational indicators has can be extended to this degree. This follows from its very definition, which we gave in one of the previous articles. Let us briefly formulate these properties (conditions: a\u003e 0, b\u003e 0, exponents p and q are irrational numbers):

Definition 4

1.a p a q \u003d a p + q

2.a p: a q \u003d a p - q

3. (a b) p \u003d a p b p

4. (a: b) p \u003d a p: b p

5. (a p) q \u003d a p q

6.a p< 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 > b p

7.a p< a q верно, если p и q – иррациональные числа, p < q , 0 < a < 1 ; если a > 0, then a p\u003e a q.

Thus, all degrees whose exponents p and q are real numbers, provided a\u003e 0, have the same properties.

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algebra 7th grade

mathematic teacher

branch MBOUTSOSH # 1

in the village of Poletaevo I.P. Zueva

Poletaevo 2016

Topic: « Natural exponent grade properties»

TARGET

1. Repetition, generalization and systematization of the studied material on the topic "Properties of the degree with a natural indicator."
2. Testing students' knowledge on this topic.
3. Application of the acquired knowledge when performing various tasks.

TASKS

subject :

to repeat, summarize and systematize knowledge on the topic; create conditions for control (mutual control) of the assimilation of knowledge and skills;continue the formation of students' motivation to study the subject;

metasubject:

develop an operational style of thinking; promote the acquisition of communication skills by students when working together; activate their creative thinking; Pcontinue the formation of certain competencies of students, which will contribute to their effective socialization; self-education and self-education skills.

personal:

to educate culture, to promote the formation of personal qualities aimed at a benevolent, tolerant attitude towards each other, people, life; foster initiative and independence in activity; lead to an understanding of the need for the topic under study for successful preparation for the state final certification.

LESSON TYPE

generalization and systematization lesson ZUN.

Equipment: computer, projector,screen for projection, board, handout.

Software: Windows 7 OS: MS Office 2007 (application required -PowerPoint).

Preparatory stage:

presentation "Properties of the degree with a natural indicator";

handout;

grade sheet.

Structure

Organizing time. Setting the goals and objectives of the lesson - 3 minutes.

Actualization, systematization of basic knowledge - 8 minutes.

Practical part -28 minutes.

Generalization, conclusion -3 minutes.

Homework - 1 minute.

Reflection - 2 minutes.

Lesson idea

Checking the ZUN of students on this topic in an interesting and effective form.

Organization of the lesson The lesson is held in grade 7. The guys work in pairs, independently, the teacher acts as a consultant-observer.

During the classes

Organizing time:

Hello guys! Today we have an unusual game lesson. Each of you is given a great opportunity to express yourself, to show your knowledge. Perhaps during the lesson you will reveal hidden abilities in yourself that will be useful to you in the future.

Each of you has a grade sheet and cards on the table for completing tasks in them. Pick up the grade sheet, you need it so that you yourself evaluate your knowledge during the lesson. Sign it up.

So, I invite you to the lesson!

Guys, look at the screen and listen to the poem.

Slide number 1

Multiply and divide

To raise a degree to a degree ...

These properties are familiar to us.

And they are not new for a long time.

Five simple rules of these

Everyone in the class has already answered

But if you forgot the properties,

Consider an example you haven't solved!

And in order to live without troubles at school

I'll give you some practical advice:

Do you want to forget the rule?

Just try to memorize!

Answer the question:

1) What actions are mentioned in it?

2) What do you think we will talk about today in the lesson?

Thus, the topic of our tutorial:

"Properties of a natural exponent" (Slide 3).

Setting goals and objectives of the lesson

In the lesson, we will repeat, summarize and bring into the system the studied material on the topic "Properties of the degree with a natural indicator"

Let's see how you learned how to multiply and divide powers with the same bases, as well as raise a power to a power

Updating basic knowledge. Systematization of theoretical material.

1) Oral work

Let's work orally

1) Formulate the properties of the degree with a natural exponent.

2) Fill in the blanks: (Slide 4)

1)5 12 : 5 5 =5 7 2) 5 7 ∙ 5 17 = 5 24 3) 5 24 : 125= 5 21 4)(5 0 ) 2 ∙5 24 =5 24

5)5 12 ∙ 5 12 = (5 8 ) 3 6)(3 12 ) 2 = 3 24 7) 13 0 ∙ 13 64 = 13 64

3) What is the value of the expression:(Slide 5-9)

a m ∙ a n; (a m + n) a m: a n (a m-n); (a m) n; a 1; a 0.

2) Checking the theoretical part (Card number 1)

Now pick up card number 1 andfill the gaps

1) If the exponent is an even number, then the value of the degree is always _______________

2) If the exponent is an odd number, then the value of the degree coincides with the sign of ____.

3) Product of degreesa n a k \u003d a n + k
When multiplying degrees with the same bases, the base is ____________, and the exponents of the degrees are ________.

4) Private degreesa n: a k \u003d a n - k
When dividing degrees with the same bases, you need a base _____, and from the index of the dividend ____________________________.

5) Exponentiation (a n) к \u003d a nk
When raising a degree to a degree, the base must be _______, and the exponents are ______.

Checking answers. (Slides 10-13)

Main part

3) And now we open notebooks, write down the number 28.01 14g, great work

Game "Clapperboard » (Slide 14)

Complete assignments in notebooks yourself

Follow the steps: a)x11 ∙ x ∙ x2 b)x14 : x5 c) (a4 ) 3 d) (-Za)2 .

Compare the value of the expression with zero: a) (- 5)7 , b) (- 6)18 ,

at 4)11 . ( -4) 8 d) (- 5) 18 ∙ (- 5) 6 , d) - (- 4)8 .

Calculate the value of the expression:

a) -1 ∙ 3 2, b) (- 1 ∙ 3) 2 c) 1 ∙ (-3) 2, d) - (2 ∙ 3) 2, e) 1 2 ∙ (-3) 2

We check if the answer is not correct. We do one hand clap.

Calculate the number of points and enter them on the score sheet.

4) Now let's do eye exercises, relieve stress, and continue working. We closely monitor the movement of objects

Getting started! (Slide 15,16,17,18).

5) Now let's get down to the next type of our work. (Card2)

Write the answer as a degree with a base FROM and you will learn the surname and name of the great French mathematician who was the first to introduce the concept of the power of a number.

Guess the name of the scientist mathematician.

1.	FROM 5 ∙ С 3	6.	FROM 7 : FROM 5
2.	FROM 8 : FROM 6	7.	(FROM 4 ) 3 ∙ С
3,	(FROM 4 ) 3	8.	FROM 4 ∙ FROM 5 ∙ С 0
4.	FROM 5 ∙ С 3 : FROM 6	9.	FROM 16 : FROM 8
5.	FROM 14 ∙ С 8	10.	(FROM 3 ) 5

ABOUT answer: RENE DECART

R

Sh

M

YU

TO

H

AND

T

E

D

FROM 8

FROM 5

FROM 1

FROM 40

FROM 13

FROM 12

FROM 9

FROM 15

FROM 2

FROM 22

Now let's listen to the student's message about "Rene Descartes"

René Descartes was born on March 21, 1596 in the small town of La Gay in Touraine. The genus Descartes belonged to the ignorant bureaucratic nobility. Rene spent his childhood in Touraine. In 1612 Descartes finished school. He spent eight and a half years in it. Descartes did not immediately find his place in life. A nobleman by birth, after graduating from college in La Flèche, he plunges headlong into the high life of Paris, then gives up everything for the sake of science. Descartes assigned mathematics a special place in his system, he considered its principles of establishing truth as a model for other sciences. A considerable merit of Descartes was the introduction of convenient designations that have survived to this day: the Latin letters x, y, z for the unknown; a, b, c - for coefficients, for degrees. Interests of Descartes are not limited to mathematics, but include mechanics, optics, biology. In 1649 Descartes, after long hesitation, moved to Sweden. This decision turned out to be fatal for his health. Six months later, Descartes died of pneumonia.

6) Work at the blackboard:

1. Solve the equation

A) x 4 ∙ (x 5) 2 / x 20: x 8 \u003d 49

B) (t 7 ∙ t 17): (t 0 ∙ t 21) \u003d -125

2. Calculate the value of the expression:

(5-x) 2 -2x 3 + 3x 2 -4x + x-x 0

a) for x \u003d -1

b) for x \u003d 2 Independently

7) Pick up card number 3, do the test

Option 1

Option 2.

1. Perform power division 217 : 2 5 2 12 2 45 2. Write in the form of a power (x + y) (x + y) \u003d x 2 + y 2 (x + y) 2 2 (x + y) 3. Replace * degree so that the equality afive · * \u003d a 15 a 10 a 3 (a 7) 5? a) a 12 b) a 5 c) a 35 3 = 8 15 8 12 6 find the meaning of a fraction

1. Perform division of degrees 99 : 9 7 9 16 9 63 2. Write it in the form of a degree (x-y) (x-y) \u003d ... x 2 -y 2 (x-y) 2 2 (x-y) 3. Replace * degree so that the equalityb 9 * \u003d b 18 b 17 b 1 1 4. What is the value of the expression(from 6) 4? a) from 10 b) from 6 c) from 24 5. From the proposed options, choose the one that can replace * in the equality (*)3 = 5 24 5 21 6 find the meaning of a fraction

Check each other's work and put your peers on the grade sheet.

Option 1	and	b	b	from	b	3
Option 2	and	b	from	from	and	4

Additional tasks for strong learners

Each assignment is assessed separately.

Find the value of an expression:

8) Now let's see the effectiveness of our lesson ( Slide 19)

To do this, completing the task, cross out the letters corresponding to the answers.

AOVSTLKRICHGNMO

Simplify the expression:

1.	С 4 ∙ С 3	5.	(FROM 2 ) 3 ∙ FROM 5
2.	(C 5) 3	6.	FROM 6 ∙ FROM 5 : FROM 10
3.	C 11: C 6	7.	(FROM 4 ) 3 ∙ С 2
4.	С 5 ∙ С 5: С

Cipher: AND - From 7 IN- From 15 G - FROM And - From 30 K - S 9 M - From 14 H - S 13 ABOUT - From 12 R - S 11 FROM - S 5 T - C 8 H - C 3

What word did you get? ANSWER: EXCELLENT! (Slide 20)

Summing up, grading, grading (Slide 21)

Let's summarize our lesson, how successfully we repeated, summarized and systematized knowledge on the topic "Properties of a degree with a natural indicator"

We take the grade sheets and calculate the total number of points and write them down in the final grade line

Stand up who scored 29-32 points: the score is excellent

25-28 points: assessment is good

20-24 points: assessment - satisfactory

Once again, I will check the correctness of the tasks on the cards, check your results with the points in the test sheet. I will put the marks in the journal

And for active work in the assessment lesson:

Guys, I ask you to evaluate your activities in the lesson. Mark in the mood sheet.

Grade sheet
Surname First name		Assessment
1.Theoretical part
2. Game "Clapperboard"
3. Test
4. "Code"
Additional part
Final grade:
Emotional assessment	About myself	About the lesson
Satisfied
Dissatisfied

Homework (Slide 22)

Create a crossword puzzle with the keyword DEGREE. In the next lesson, we will look at the most interesting works.

№ 567

List of sources used

1. Textbook "Algebra Grade 7".
2. Poem. http://yandex.ru/yandsearch
3. NOT. Shchurkov. The culture of the modern lesson. Moscow: Russian Pedagogical Agency, 1997.
4. A.V. Petrov. Methodological and methodological foundations of personality-developing computer education. Volgograd. Change, 2001.
5. A.S. Belkin. Success situation. How to create it. M .: "Education", 1991.
6. Informatics and Education # 3. Operational thinking style, 2003

Lesson flow chart

Grade 7 Lesson number 38

Topic: Degree with natural exponent

1. Provide repetition, generalization and systematization of knowledge on the topic, consolidate and improve the skills of the simplest transformations of expressions containing degrees with a natural indicator, create conditions for controlling the assimilation of knowledge and skills;

2. To promote the formation of skills to apply the techniques of generalization, comparison, highlighting the main thing, to promote the education of interest in transferring knowledge to a new situation, the development of mathematical horizons, speech, attention and memory, the development of educational and cognitive activities;

3. To promote the fostering of interest in mathematics, activity, organization, to foster the skills of mutual and self-control of their activities, the formation of positive motivation for learning, a culture of communication.

Basic concepts of the lesson

Degree, base of degree, exponent, properties of degree, product of degree, division of degrees, raising a degree to a power.

Planned result

Learn to operate with the concept of Degree, understand the meaning of writing a number in the form of a degree, and perform simple transformations of expressions containing degrees with a natural exponent.

They will be able to learn how to perform transformations of integer expressions containing a degree with a natural exponent

Subject skills, UUD

Personal UUD:

the ability to self-esteem based on the criterion of the success of educational activities.

Cognitive UUD:

the ability to navigate in your system of knowledge and skills: to distinguish new from already known with the help of a teacher; find answers to questions using the information learned in the lesson.

Generalization and systematization of educational material, operate with a symbolic recording of the degree, substitutions, reproduce from memory the information necessary for solving the educational problem

Subject UUD:

Apply degree properties to transform expressions containing natural exponents

Regulatory UUD:

Ability to define and formulate a goal in the lesson with the help of a teacher; evaluate your work in the lesson.Exercise mutual control and self-control when performing tasks

Communicative UUD:
Be able to formulate your thoughts orally and in writing, listen and understand the speech of others

Metasubject links

Physics, astronomy, medicine, everyday life

Lesson type

Repetition, generalization and application of knowledge and skills.

Forms of work and methods of work

Frontal, steam room, individual. Explanatory - illustrative, verbal, problem situation, workshop, mutual check, control

Resource provision

Components of EMC Makarychev Textbook, projector, screen, computer, presentation, assignments for students, self-assessment sheets

Technologies used in the classroom

Semantic reading technology, problem learning, individual and differentiated approach, ICT

Mobilizing students to work, mobilizing attention

Good afternoon guys. Good afternoon, dear colleagues! I greet everyone gathered for tonight open lesson... Guys, I want to wish you fruitful work in the lesson, carefully consider the answers to the questions posed, not rush, not interrupt, respect classmates and their answers. And I also wish you all get only good grades. Good luck to you!

Are included in the business rhythm of the lesson

They check the availability of everything necessary for work in the lesson, the accuracy of the arrangement of objects. Ability to organize oneself, tune in to work.

2. Actualization of basic knowledge and entry into the topic of the lesson

3. Oral work

Guys, each of you has score sheets on your desk.On them you will evaluate your work in the lesson. Today in the lesson you are given the opportunity to receive not one, but two marks: for work in the lesson and for independent work.
Your correct, complete answers will also be rated "+", but I will put this assessment in another column.

On the screen, you see puzzles in which the keywords of today's lesson are encrypted. Unravel them. (Slide 1)

power

reiteration

generalization

Guys, you guessed the puzzles correctly. These words are: degree, repetition and generalization. Now, using the guessed words - clues, formulate the topic of today's lesson.

Right. Open notebooks and write down the number and the topic of the lesson "Repetition and generalization on the topic" Properties of the degree with a natural indicator "(Slide 2)

We have identified the topic of the lesson, but what do you think we will do in the lesson, what goals will we set for ourselves? (Slide 3)

Repeat and generalize our knowledge on this topic, fill in the existing gaps, prepare for the study of the next topic "Monomials".

Guys, the properties of the degree with a natural exponent are quite often used when finding the values \u200b\u200bof expressions, when converting expressions. The speed of calculations and transformations related to the properties of a degree with a natural exponent is also dictated by the introduction of the USE.

So, today we will review and summarize your knowledge and skills on this topic. Verbally, you must solve a number of problems and remember the verbal grouping of properties and determination of the degree with a natural indicator.

Epigraph to the lesson words of the great Russian scientist MV Lomonosov "Let someone try to delete degrees from mathematics, and he will see that without them you cannot go far"

(Slide 4)

Do you think the scientist is right?

Why do we need degrees?

Where are they widely used? (in physics, astronomy, medicine)

That's right, now let's repeat, what is a degree?

What are the names of a andn in writing the degree?

What actions can you do with degrees? (Slides 5-11)

Now let's summarize. You have assignment sheets on your desk .

1. On the left are the beginning of the definitions on the right, the end of the definitions. Connect with lines the correct statements (Slide 12)

Connect the corresponding parts of the definition with lines.

a) When multiplying degrees with the same bases ...

1) the basis of the degree

b) When dividing degrees with the same bases….

2) Exponent

c) The number a is called

3) the product of n factors, each of which is equal to a.

d) When raising a degree to a degree ...

4)… the base remains the same, and the indicators add up.

e) The power of a number a with a natural exponent n greater than 1 is called

5) ... the base remains the same, and the indicators are multiplied.

e)Numberncalled

6) Degree

g)Expression a n called

7)… the basis remains the same and the values \u200b\u200bare deducted.

2.Now, swap papers with your deskmate, rate his work and give him a grade. Put this grade on your scorecard.

Now let's check if you completed the task correctly.

Guess puzzles, define words - clues.

Attempts are made to present the topic of the lesson.

Write down the number and topic of the lesson in a notebook.

Answer questions

They work in pairs. They read the assignment, remember.

Connect parts of definitions

Exchanging notebooks.

They carry out a mutual check of the results, give grades to a neighbor on a desk.

4.Physical training

Hands raised and shook -

these are the trees in the forest,

Arms bent, brushes shook -

The wind tears off the foliage.

To the sides of the hand, gently wave -

Birds fly south so

We will quietly show how they sit down -

Hands folded like this!

Perform actions in parallel with the teacher

5. Transfer of acquired knowledge, their primary application in new or changed conditions, in order to form skills.

1. I offer you the following job: you have cards on your desks. You need to complete tasks i.e. write the answer in the form of a degree with a base s, and you will learn the surname and name of the great French mathematician who introduced the currently accepted designation of degrees. (Slide 14)

5

FROM 8 : FROM 6

(FROM 4 ) 3 FROM

(FROM 4 ) 3

FROM 4 FROM 5 FROM 0

FROM 5 FROM 3 : FROM 6

FROM 16 : FROM 8

FROM 14 FROM 8

10.

(FROM 3 ) 5

Answer: Rene Descartes.

A story about the biography of Rene Descartes (Slides 15 - 17)

Guys, now let's do the next task.

2. About limit which answers are correct and which are false. (Slide 18 - 19)

match the true answer with 1, and the false one with 0.

having received an ordered set of ones and zeros, you will find out the correct answer and determine the first and last name of the first Russian woman - a mathematician.

and) x 2 x 3 \u003d x 5

b) s 3 s 5 s 8 = s 16

in) x 7 : x 4 \u003d x 28

d) (c+ d) 8 : ( c+ d) 7 = c+ d

e) (x 5 ) 6 = x 30

Choose her name from four names of famous women, each of which corresponds to a set of ones and zeros:

Ada Augusta Lovelace - 11001

Sophie Germain - 10101

Ekaterina Dashkova - 11101

Sofia Kovalevskaya - 11011

From the biography of Sophia Kovalevskaya (Slide 20)

Complete the task, determine the surname and first name of the French mathematician

Listen, consider slides

They mark the correct and incorrect answers, write down the resulting code, which determines the name of the first Russian woman - a mathematician.

6. Control and assessment of knowledge Students' independent fulfillment of assignments under the supervision of a teacher.

And now you have to do the verification work. Before you are cards with tasks of different colors. The color corresponds to the level of difficulty of the assignment (at "3", at "4", at "5") Choose for yourself, the assignment for which grade you will perform and get to work. (Slide 21)

On "3"

1. Imagine the work as a degree:

and) ; b) ;

in) ; d) .

2. Follow the steps:

( m 3 ) 7 ; ( k 4 ) 5 ; (2 2 ) 3; (3 2 ) 5 ; ( m 3 ) 2 ; ( a x ) y

On "4"

1. Present the work as a degree.

a) x 5 x 8 ; boo 2 at 9 ; at 2 6 · 2 4 ; d)m 2 m 5 m 4 ;

e)x 6 ∙ x 3 ∙ x 7 ; e) (–7) 3 ∙ (–7) 2 ∙ (–7) 9 .

2. Imagine the quotient as a degree:

and)x 8 : x 4 ; b) (–0.5) 10 : (–0,5) 8 ;

c) x 5 : x 3 ; d) at 10 : at 10 ; D 2 6 : 2 4 ; e);

to "5"

1. Follow the steps:

a) a 4 · and · and 3 a b) (7 x ) 2 c) p · r 2 · r 0

d) with · from 3 · c e) t · t 4 · ( t 2 ) 2 · t 0

e) (2 3 ) 7 : (2 5 ) 3 g) -x 3 · (– x ) 4

h) (r 2 ) 4 : r 5 and) (3 4 ) 2 · (3 2 ) 3 : 3 11

2. Simplify:

and) x 3 ( x 2) 5 c) ( a 2) 3 ( a 4 ) 2

b) ( a 3) 2 a 5 g) ( x 2) 5 ( x 5 )

Independent work

Perform assignments in notebooks

7. Lesson summary

Generalization of the information received in the lesson.Checking work, assigning marks. Identification of difficulties encountered in the lesson

8. Reflection

What happened to the concept of a degree inXVII century, you and I can predict ourselves. To do this, try to answer the question: can a number be raised to a negative power or fractional? But this is the subject of our future study.

Lesson grades

Guys, I want to finish our lesson with the following parable.

Parable. A wise man was walking, and three people met him, who were carrying carts with stones for construction under the hot sun. The sage stopped and asked each one a question. The first one asked: “What have you been doing all day”. And he answered with a grin that he had been driving the damned stones all day. The sage asked the second: "What have you been doing all day," and he replied: "But I did my job conscientiously." And the third smiled, his face lit up with joy and pleasure: "And I took part in the construction of the temple!"

Guys, answer, what did you do in class today? Just do it on your self-assessment sheet. Circle the statement that applies to you in each column.

On the self-assessment sheet, you need to emphasize phrases that characterize the student's work in the lesson in three areas.

Our lesson is over. Thank you all for the work in the lesson!

Answer questions

Assess their work in the classroom.

They mark phrases in the card that characterize their work in the lesson.