How to square root. Engineering calculator. Entering under the sign of the root

Degree formulas used in the process of reduction and simplification complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:

Operations with degrees.

1. By multiplying degrees with the same base, their indicators are added:

a m·a n = a m + n .

2. When dividing degrees with the same base, their exponents are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n · b n · c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n /b n .

5. Raising a power to a power, the exponents are multiplied:

(a m) n = a m n .

Each formula above is true in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the radical number to this power:

4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:

5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:

A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to absolute value non-positive indicator:

Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.

A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.

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

A fairly 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” button (“Insert Function”), which is located 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 “Mathematical” category or in the “Full alphabetical list” by scrolling the menu just below with the mouse.

  4. Select the “Root” item by clicking once with the left mouse button, then click the “OK” button.

  5. The following menu will appear - “Function Arguments”.

  6. Enter a number or select a cell in which this expression or formula was previously written, to do 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 entered into the line.

  7. Click on the "OK" button.

  8. And everything is ready, the function has calculated 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 (the data that is entered in a given cell) or two cells; to do this, enter the values ​​in the “Number” line. Write the number and click once on the cell, the program itself will add an addition sign.

On a note! This function can also be entered manually. In the formula bar, enter the following expression: “=ROOT(x)”, where x is the required number.

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

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

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

For example:

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

There are two ways to perform this action in a spreadsheet program:

  1. Using a function.
  2. Using the power symbol "^", enter the expression manually.

Extracting a 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 in the “Category” item, in the “Mathematical” or “Full alphabetical list” category, find the “Degree” function.

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

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

    Important! To indicate a division sign, you must use the "/" symbol rather than the standard ":" division sign.

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

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

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


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

Examples of writing expressions


From this article you will learn:

  • what is “root extraction”;
  • in what cases is it removed;
  • principles of finding the root value;
  • basic methods of extracting roots from natural and fractional numbers.

What is "root extraction"

First, let’s introduce the definition of “root extraction.”

Definition 1

Root extraction is the process of finding the value of the root.

When we take the nth root of a number, we find the number b, the nth power of which is equal to a. If we find such a number b, we can say that the root has been extracted.

Note 1

The expressions “extracting the root” and “finding the value of the root” are equivalent.

In what cases is the root extracted?

Definition 2

The nth root can be extracted from a number exactly if a can be represented as the nth power of some number b.

Example 1

4 = 2 × 2, therefore, the square root of the number 4 can be exactly taken, which is 2

Definition 3

When the n-th root of a number cannot be represented as the n-th power of b, then such a root not extracted or only approximate value is retrieved root accurate to any decimal place.

Example 2

2 ≈ 1 , 4142 .

Principles of finding root values ​​and methods of extracting them

  • Using a table of squares, a table of cubes, etc.
  • Decomposition of a radical expression (number) into prime factors
  • Taking the root of a negative number

It is necessary to understand by what principles the meaning of the roots is found and how they are extracted.

Definition 4

The main principle of finding the value of the roots is to be based on the properties of the roots, including the equality: b n n = b, which is valid for any non-negative number b.

You should start with the simplest and most obvious method: tables of squares, cubes, etc.

When you don’t have a table at hand, the method of decomposing a radical number into prime factors will help you (the method is simple).

It is worth paying attention to extracting the root of a negative number, which is possible for roots with odd exponents.

Let's learn how to take roots from fractions, including mixed numbers, fractions, and decimals.

And we will slowly consider the method of finding the value of the root bit by bit - the most complex and multi-stage one.

Using a table of squares, cubes, etc.

The table of squares includes all numbers from 0 to 99 and consists of 2 zones: in the first zone you can make any number up to 99 using a vertical column with tens and a horizontal row with units, the second zone contains all the squares of the numbers formed.

Table of squares

Table of squares units
0 1 2 3 4 5 6 7 8 9
tens 0 0 1 4 9 16 25 36 49 64 81
1 100 121 144 169 196 225 256 289 324 361
2 400 441 484 529 576 625 676 729 784 841
3 900 961 1024 1089 1156 1225 1296 1369 1444 1521
4 1600 1681 1764 1849 1936 2025 2116 2209 2304 2041
5 2500 2601 2704 2809 2916 3025 3136 3249 3364 3481
6 3600 3721 3844 3969 4096 4225 4356 4489 4624 4761
7 4900 5041 5184 5329 5476 5625 5776 5929 6084 6241
8 6400 6561 6724 6889 7056 7225 7396 7569 7744 7921
9 8100 8281 8464 8649 8836 9025 9216 9409 9604 9801

There are also tables of cubes, fourth powers, etc., which are created on a principle similar to the table of squares.

Cube table

Cube table units
0 1 2 3 4 5 6 7 8 9
tens 0 0 1 8 27 64 125 216 343 512 729
1 1000 1 331 1 728 2 197 2 744 3 375 4 096 4 913 5 832 6 859
2 8000 9 261 10 648 12 167 13 824 15 625 17 576 19 683 21 952 24 389
3 27000 29 791 32 768 35 937 39 304 42 875 46 656 50 653 54 872 59 319
4 64000 68 921 74 088 79 507 85 184 91 125 97 336 103 823 110 592 117 649
5 125000 132 651 140 608 148 877 157 464 166 375 175 616 185 193 195 112 205 379
6 216000 226 981 238 328 250 047 262 144 274 625 287 496 300 763 314 432 328 509
7 343000 357 911 373 248 389 017 405 224 421 875 438 976 456 533 474 552 493 039
8 512000 531 441 551 368 571 787 592 704 614 125 636 056 658 503 681 472 704 969
729000 753 571 778 688 804 357 830 584 857 375 884 736 912 673 941 192 970 299

The principle of operation of such tables is simple, but they are often not at hand, which greatly complicates the root extraction process, so you need to know at least several methods of root extraction.

Factoring a radical number into prime factors

The most convenient way to find the root value after a table of squares and cubes.

Definition 5

The method of decomposing a radical number into prime factors involves representing the number as a power with the necessary exponent, which allows us to obtain the value of the root.

Example 3

Let's take the square root of 144.

Let's factor 144 into prime factors:

Thus: 144 = 2 × 2 × 2 × 2 × 3 × 3 = (2 × 2) 2 × 3 2 = (2 × 2 × 3) 2 = 12 2. Therefore, 144 = 12 2 = 12.

Also, when using the properties of powers and roots, you can write the transformation a little differently:

144 = 2 × 2 × 2 × 2 × 3 × 3 = 2 4 × 3 2 = 2 4 × 3 2 = 2 2 × 3 = 12

144 = 12 is the final answer.

Extracting roots from fractional numbers

Let's remember: any fractional number must be written as common fraction.

Definition 6

Following the property of the root of a quotient, the following equality is valid:

p q n = p n q n . Based on this equality, it is necessary to use rule for extracting the root of a fraction: The root of a fraction is equal to the root of the numerator divided by the root of the denominator.

Example 4

Let's consider an example of extracting a root from a decimal fraction, since you can extract a root from an ordinary fraction using a table.

It is necessary to extract the cube root of 474, 552. First of all, let's imagine the decimal fraction as an ordinary fraction: 474, 552 = 474552 / 1000. From this it follows: 474552 1000 3 = 474552 3 1000 3. You can then begin the process of extracting the cube roots of the numerator and denominator:

474552 = 2 × 2 × 2 × 3 × 3 × 3 × 13 × 13 × 13 = (2 × 3 × 13) 3 = 78 3 and 1000 = 10 3, then

474552 3 = 78 3 3 = 78 and 1000 3 = 10 3 3 = 10.

We complete the calculations: 474552 3 1000 3 = 78 10 = 7, 8.

Rooting Negative Numbers

If the denominator is an odd number, then the number under the root sign may be negative. It follows from this: for a negative number - a and an odd exponent of the root 2 n - 1, the following equality holds:

A 2 × n - 1 = - a 2 × n - 1

Definition 7

Rule for extracting odd powers from negative numbers: to extract the root of a negative number you need to extract the root of its opposite positive number and put a minus sign in front of it.

Example 5

12 209 243 5. First, you need to transform the expression so that there is a positive number under the root sign:

12 209 243 5 = 12 209 243 - 5 ​​​​​​

Then it should be replaced mixed number ordinary fraction:

12 209 243 - 5 = 3125 243 - 5

Using the rule for extracting roots from an ordinary fraction, we extract:

3125 243 - 5 = - 3125 5 243 5

We calculate the roots in the numerator and denominator:

3125 5 243 5 = - 5 5 5 3 5 5 = - 5 3 = - 1 2 3

Brief summary of the solution:

12 209 243 5 = 12 209 243 - 5 = 3125 243 - 5 = - 3125 5 243 5 = - 5 5 5 3 5 5 = - 5 3 = - 1 2 3 .

Answer: - 12 209 243 5 = - 1 2 3.

Bitwise determination of the root value

There are cases when under the root there is a number that cannot be represented as the nth power of a certain number. But it is necessary to know the value of the root accurate to a certain sign.

In this case, it is necessary to use an algorithm for finding the value of the root bitwise, with the help of which you can obtain a sufficient number of values ​​of the desired number.

Example 6

Let's look at how this happens using the example of extracting the square root of 5.

First you need to find the value of the units digit. To do this, let's start going through the values ​​0, 1, 2, . . . , 9 , while calculating 0 2 , 1 2 , . . . , 9 2 to the required value, which is greater than the radical number 5. It is convenient to present all this in the form of a table:

The value of a row of units is 2 (since 2 2< 5 , а 2 3 >5) . Let's move to the category of tenths - we will square the numbers 2, 0, 2, 1, 2, 2, . . . , 2, 9, comparing the obtained values ​​with the number 5.

Since 2, 2 2< 5 , а 2 , 3 2 >5, then the value of the tenths is 2. Let's move on to finding the value of hundredths:

Thus, the value of the root of five is found - 2, 23. You can find the root values ​​further:

2 , 236 , 2 , 2360 , 2 , 23606 , 2 , 236067 , . . .

So, we have studied several of the most common ways to find the value of the root, which can be used in any situation.

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The nth root of a number x is a non-negative number z that, when raised to the nth power, becomes x. Determining the root is included in the list of basic arithmetic operations that we become familiar with in childhood.

Mathematical notation

"Root" comes from the Latin word radix and today the word "radical" is used as a synonym for this mathematical term. Since the 13th century, mathematicians have denoted the root operation by the letter r with a horizontal bar over the radical expression. In the 16th century, the designation V was introduced, which gradually replaced the sign r, but the horizontal line remained. It is easy to type in a printing house or write by hand, but in electronic publishing and programming the letter designation of the root has spread - sqrt. This is how we will denote square roots in this article.

Square root

The square radical of a number x is a number z that, when multiplied by itself, becomes x. For example, if we multiply 2 by 2, we get 4. Two in this case is the square root of four. Multiply 5 by 5, we get 25 and now we already know the value of the expression sqrt(25). We can multiply and – 12 by −12 to get 144, and the radical of 144 is both 12 and −12. Obviously, square roots can be both positive and negative numbers.

The peculiar dualism of such roots is important for solving quadratic equations, therefore, when searching for answers in such problems, you need to indicate both roots. When deciding algebraic expressions Arithmetic square roots are used, that is, only their positive values.

Numbers whose square roots are integers are called perfect squares. There is a whole sequence of such numbers, the beginning of which looks like:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256…

The square roots of other numbers are irrational numbers. For example, sqrt(3) = 1.73205080757... and so on. This number is infinite and non-periodic, which causes some difficulties in calculating such radicals.

The school mathematics course states that you cannot take square roots of negative numbers. As we learn in a university course on mathematical analysis, this can and should be done - this is why complex numbers are needed. However, our program is designed to extract real root values, so it does not calculate even radicals from negative numbers.

Cube root

The cubic radical of a number x is a number z that, when multiplied by itself three times, gives the number x. For example, if we multiply 2 × 2 × 2, we get 8. Therefore, two is the cube root of eight. Multiply the four by itself three times and get 4 × 4 × 4 = 64. Obviously, the four is the cube root of the number 64. There is an infinite sequence of numbers whose cubic radicals are integers. Its beginning looks like:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744…

For other numbers, the cube roots are irrational numbers. Unlike square radicals, cube roots, like any odd roots, can be derived from negative numbers. It's all about the product of numbers less than zero. Minus for minus gives a plus - a rule known from school. And a minus for a plus gives a minus. If we multiply negative numbers an odd number of times, the result will also be negative, therefore, nothing prevents us from extracting an odd radical from a negative number.

However, the calculator program works differently. Essentially, extracting a root is raising it to the inverse power. The square root is considered to be raised to the power of 1/2, and the cubic root is considered to be raised to the power of 1/3. The formula for raising to the power of 1/3 can be rearranged and expressed as 2/6. The result is the same, but you cannot extract such a root from a negative number. Thus, our calculator calculates arithmetic roots only from positive numbers.

nth root

Such an ornate method of calculating radicals allows you to determine roots of any degree from any expression. You can take the fifth root of a cube of a number or the 19th radical of a number to the 12th power. All this is elegantly implemented in the form of raising to the power of 3/5 or 12/19, respectively.

Let's look at an example

Diagonal of a square

The irrationality of the diagonal of a square was known to the ancient Greeks. They were faced with the problem of calculating the diagonal of a flat square, since its length is always proportional to the root of two. The formula for determining the length of the diagonal is derived from and ultimately takes the form:

d = a × sqrt(2).

Let's determine the square radical of two using our calculator. Let’s enter the value 2 in the “Number(x)” cell, and also 2 in the “Degree(n)” cell. As a result, we get the expression sqrt(2) = 1.4142. Thus, to roughly estimate the diagonal of a square, it is enough to multiply its side by 1.4142.

Conclusion

Finding a radical is a standard arithmetic operation, without which scientific or design calculations are indispensable. Of course, we don’t need to determine roots to solve everyday problems, but our online calculator will definitely be useful for schoolchildren or students to check homework in algebra or calculus.