How to calculate the root of a number to a power. Engineering calculator. Introduction under the root sign

To successfully use the operation of extracting a root in practice, you need to become familiar with the properties of this operation.
All properties are formulated and proved only for non-negative values ​​of the variables contained under the signs of the roots.

Theorem 1. Root nth degree(n = 2, 3, 4, ...) from the product of two non-negative chipcell is equal to the product roots of the nth powers of these numbers:

Comment:

1. Theorem 1 remains valid for the case when the radical expression is the product of more than two non-negative numbers.

Theorem 2.If, and n - natural number greater than 1, then the equality

Brief(albeit imprecise) formulation that is more convenient to use in practice: the root of a fraction is equal to the fraction of the roots.

Theorem 1 allows us to multiply m only roots of the same degree , i.e. only roots with the same index.

Theorem 3 If ,k is a natural number and n is a natural number greater than 1, then the equality

In other words, to build a root in natural degree, it is enough to raise the radical expression to this degree.
This is a consequence of Theorem 1. Indeed, for example, for k = 3 we obtain: In the same way, one can reason in the case of any other natural value of the exponent k.

Theorem 4 If ,k, n are natural numbers greater than 1, then the equality

In other words, to extract a root from a root, it is enough to multiply the indices of the roots.
For instance,

Be careful! We learned that four operations can be performed on roots: multiplication, division, exponentiation, and root extraction (from the root). But what about the addition and subtraction of roots? No way.
For example, instead of it is impossible to write Indeed, But it is obvious that

Theorem 5 If the indices of the root and the radical expression are multiplied or divided by the same natural number, then the value of the root will not change, i.e.

Examples of solving tasks

Example 1. Calculate

Solution. Using the first property of roots (Theorem 1), we get:

Example 2. Calculate
Solution. Convert the mixed number to an improper fraction.
We have Using the second property of the roots ( Theorem 2 ), we get:

Example 3. Calculate:

Solution. Any formula in algebra, as you well know, is used not only "from left to right", but also "from right to left". So, the first property of roots means that it can be represented in the form and, conversely, can be replaced by an expression. The same applies to the second property of roots. With this in mind, let's perform the calculations.

Examples:

\ (\ sqrt (16) = 2 \) since \ (2 ^ 4 = 16 \)
\ (\ sqrt (- \ frac (1) (125)) \) \ (= \) \ (- \ frac (1) (5) \), because \ ((- \ frac (1) (5) ) ^ 3 \) \ (= \) \ (- \ frac (1) (125) \)

How to calculate the nth root?

To calculate the root of the \ (n \) - th degree, you need to ask yourself the question: what number in the \ (n \) - th power will give under the root?

for instance... Calculate the root \ (n \) - th degree: a) \ (\ sqrt (16) \); b) \ (\ sqrt (-64) \); c) \ (\ sqrt (0.00001) \); d) \ (\ sqrt (8000) \); e) \ (\ sqrt (\ frac (1) (81)) \).

a) What number in the \ (4 \) - th degree will give \ (16 \)? Obviously, \ (2 \). So:

b) What number in the \ (3 \) -th degree will give \ (- 64 \)?

\ (\ sqrt (-64) = - 4 \)

c) What number in the \ (5 \) - th degree will give \ (0.00001 \)?

\ (\ sqrt (0.00001) = 0.1 \)

d) What number in the \ (3 \) -th degree will give \ (8000 \)?

\ (\ sqrt (8000) = 20 \)

e) What number in the \ (4 \) - th degree will \ (\ frac (1) (81) \) give?

\ (\ sqrt (\ frac (1) (81)) = \ frac (1) (3) \)

We have considered the most simple examples with the root \ (n \) - th degree. To solve more complex problems with roots \ (n \) - th degree - it is vital to know them.

Example. Calculate:

\ (\ sqrt 3 \ cdot \ sqrt (-3) \ cdot \ sqrt (27) \ cdot \ sqrt (9) - \) \ (= \)		At the moment, none of the roots can be calculated. Therefore, we apply the properties of the root \ (n \) - th degree and transform the expression. \ (\ frac (\ sqrt (-64)) (\ sqrt (2)) \)\ (= \) \ (\ sqrt (\ frac (-64) (2)) \) \ (= \) \ (\ sqrt (-32) \) because \ (\ frac (\ sqrt [n] (a)) (\ sqrt [n] (b)) \)\ (= \) \ (\ sqrt [n] (\ frac (a) (b)) \)
\ (= \ sqrt (3) \ cdot \ sqrt (-3) \ cdot \ sqrt (27) \ cdot \ sqrt (9) - \ sqrt (-32) = \)		Let us rearrange the factors in the first term so that Square root and the root \ (n \) - th degree stood side by side. This will make it easier to apply properties as most of the properties of \ (n \) -th roots work only with roots of the same degree. And we calculate the root of the 5th degree.
\ (= \ sqrt (3) \ cdot \ sqrt (27) \ cdot \ sqrt (-3) \ cdot \ sqrt (9) - (- 5) = \)		Apply the property \ (\ sqrt [n] (a) \ cdot \ sqrt [n] (b) = \ sqrt [n] (a \ cdot b) \) and expand the bracket
\ (= \ sqrt (81) \ cdot \ sqrt (-27) + 5 = \)		Calculate \ (\ sqrt (81) \) and \ (\ sqrt (-27) \)
\ (= 9 \ cdot (-3) +5 = -27 + 5 = -22 \)

Are nth root and square root related?

In any case, any root of any degree is just a number, albeit written in an unfamiliar form.

Feature of the root of the n-th degree

The root \ (n \) - th power with odd \ (n \) can be extracted from any number, even negative (see examples at the beginning). But if \ (n \) is even (\ (\ sqrt (a) \), \ (\ sqrt (a) \), \ (\ sqrt (a) \) ...), then such a root is extracted only if \ ( a ≥ 0 \) (by the way, the square root has the same). This is because extracting a root is the opposite of exponentiation.

And raising to an even power makes even a negative number positive. Indeed, \ ((- 2) ^ 6 = (- 2) \ cdot (-2) \ cdot (-2) \ cdot (-2) \ cdot (-2) \ cdot (-2) = 64 \). Therefore, we cannot get an even power of a negative number under the root. This means that we cannot extract such a root from a negative number.

The odd degree of such restrictions does not have - a negative number raised to an odd degree will remain negative: \ ((- 2) ^ 5 = (- 2) \ cdot (-2) \ cdot (-2) \ cdot (-2) \ cdot (-2) = - 32 \). Therefore, under the root of an odd degree, you can get a negative number. This means that you can also extract it from a negative number.

Engineering calculator online

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Undoubtedly, Web20calc will be of interest to that group of people who, in search of simple solutions, types a query in search engines: mathematical online calculator... A free web application will help you instantly calculate the result of some mathematical expression, for example, subtract, add, divide, extract a root, raise to a power, etc.

In the expression, you can use the operations of exponentiation, addition, subtraction, multiplication, division, percentage, constant PI. For complex calculations, use parentheses.

Engineering calculator features:

1. basic arithmetic operations;
2. work with numbers in a standard form;
3. calculation of trigonometric roots, functions, logarithms, exponentiation;
4. statistical calculations: addition, arithmetic mean or standard deviation;
5. application of a memory cell and user-defined functions of 2 variables;
6. work with angles in radian and degree measures.

The engineering calculator allows you to use a variety of mathematical functions:

Extraction of roots (square root, cubic, and n-th root);
ex (e to the x power), exponent;
trigonometric functions: sine - sin, cosine - cos, tangent - tan;
inverse trigonometric functions: arcsine - sin-1, arccosine - cos-1, arctangent - tan-1;
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To make mathematical calculations, first enter a sequence of mathematical expressions in the appropriate field, then click on the equal sign and see the result. You can enter values ​​directly from the keyboard (for this, the calculator area must be active, therefore, it will not be superfluous to put the cursor in the input field). Among other things, data can be entered using the buttons on the calculator itself.

To build graphs in the input field, write the function as indicated in the field with examples or use the specially designed toolbar (to go to it, click on the button with the icon in the form of a graph). To convert values ​​press Unit, to work with matrices - Matrix.

Spreadsheet users make extensive use of the function to extract the root of a number. Since working with data usually requires processing large numbers, manual counting can be quite difficult. In this article, you will find a detailed analysis of the question of extracting root of any degree in Excel.

Quite an easy task, since the program has a separate function that can be taken from the list. To do this, you need to do the following:

1. Select the cell in which you want to register the function by clicking on it once with the left mouse button. A black outline appears, the active row and column are highlighted in orange, and the name appears in the address cell.

   

2. Click on the "fx" (Insert Function) button above the column names, after the address cell, before the formula bar.

   

3. A drop-down menu will appear in which you need to find the "Root" function. This can be done in the "Maths" category or in the "Complete alphabetical list" by scrolling down the menu with the mouse.

   

4. Select the "Root" item by clicking once with the left mouse button, then - the "OK" button.

   

5. The following menu appears - "Function arguments".

   

6. Enter a number or select a cell in which this expression or formula was previously written, for this, left-click once on the "Number" line, then move the cursor over the cell you need and click on it. The cell name will be automatically filled into the string.

   

7. Click on the "OK" button.

   

8. And everything is ready, the function calculated the square root, writing the result to the selected cell.

   

It is also possible to extract the square root of the sum of a number and a cell (data that is packed in this cell) or two cells, for this enter the values ​​in the "Number" line. Write the number and click once on the cell, the program will put the addition sign itself.

On a note! This function can also be entered manually. In the formula bar, enter the following expression: "= ROOT (x)", where x is the number you are looking for.

Extraction of roots of 3rd, 4th and other degrees.

There is no separate function for solving this expression in Excel. To extract the n-th root, you must first consider it from a mathematical point of view.

The nth root is equal to raising a number to the opposite power (1 / n). That is, the square root is the ½ (or 0.5) power.

For instance:

• the fourth root of 16 is 16 to the power ¼;
• cube root of 64 = 64 to the 1/3 power;

There are two ways to do this in a spreadsheet program:

1. Using the function.
2. Using the degree icon “^”, enter the expression manually.

Extracting root of any degree using a function

1. Select the desired cell and click on "Insert Function" in the "Formulas" tab.

   

2. Expand the list under Category, under Math or Complete Alphabetical List, find the Degree function.

   

   

3. In the "Number" line, enter a number (in our case, this is the number 64) or the name of a cell by clicking on it once.

   

4. In the line "Degree" type the degree to which you want to raise the root (1/3).

   

   Important! To indicate the division sign, you must use the "/" sign, and not the standard division sign ":".

5. Click "OK" and the result of the action will appear in the originally selected cell.

   

   

Note! For the most detailed instructions with a photo on working with functions, see the article above.

Extract root of any degree using the degree symbol "^"

Note! The degree can be written either as a fraction or decimal number... For example, the fraction ¼ can be written as 0.25. To separate tenths, hundredths, thousandths, and so on, use a comma, as is customary in mathematics.

Examples of writing expressions