Before the advent of calculators, students and teachers calculated square roots by hand. There are several ways to manually calculate the square root of a number. Some of them offer only an approximate solution, others give an exact answer.

Steps

Prime factorization

Factor the root number into factors that are square numbers. Depending on the root number, you will get an approximate or exact answer. Square numbers are numbers from which the whole square root can be taken. Factors are numbers that, when multiplied, give the original number. For example, the factors of the number 8 are 2 and 4, since 2 x 4 = 8, the numbers 25, 36, 49 are square numbers, since √25 = 5, √36 = 6, √49 = 7. Square factors are factors , which are square numbers. First, try to factorize the root number into square factors.

• For example, calculate the square root of 400 (manually). First try factoring 400 into square factors. 400 is a multiple of 100, that is, divisible by 25 - this is a square number. Dividing 400 by 25 gives you 16. The number 16 is also a square number. Thus, 400 can be factored into square factors of 25 and 16, that is, 25 x 16 = 400.
• This can be written as follows: √400 = √(25 x 16).

1. The square root of the product of some terms is equal to the product of the square roots of each term, that is, √(a x b) = √a x √b. Use this rule and take the square root of each square factor and multiply the results to find the answer.

   • In our example, take the square root of 25 and 16.
     • √(25 x 16)
     • √25 x √16
     • 5 x 4 = 20

2. If the radical number does not factor into two square factors (and it does in most cases), you will not be able to find the exact answer as an integer. But you can simplify the problem by decomposing the root number into a square factor and an ordinary factor (a number from which the whole square root cannot be taken). Then you will take the square root of the square factor and you will take the root of the ordinary factor.

   • For example, calculate the square root of the number 147. The number 147 cannot be factored into two square factors, but it can be factored into the following factors: 49 and 3. Solve the problem as follows:
     • = √(49 x 3)
     • = √49 x √3
     • = 7√3

3. If necessary, evaluate the value of the root. Now you can evaluate the value of the root (find an approximate value) by comparing it with the values ​​​​of the roots of square numbers that are closest (on both sides of the number line) to the root number. You will get the value of the root as a decimal fraction, which must be multiplied by the number behind the root sign.

   • Let's go back to our example. The root number is 3. The nearest square numbers to it are the numbers 1 (√1 = 1) and 4 (√4 = 2). Thus, the value of √3 lies between 1 and 2. Since the value of √3 is probably closer to 2 than to 1, our estimate is: √3 = 1.7. We multiply this value by the number at the root sign: 7 x 1.7 \u003d 11.9. If you do the calculations on a calculator, you get 12.13, which is pretty close to our answer.
     • This method also works with large numbers. For example, consider √35. The root number is 35. The nearest square numbers to it are the numbers 25 (√25 = 5) and 36 (√36 = 6). Thus, the value of √35 lies between 5 and 6. Since the value of √35 is much closer to 6 than it is to 5 (because 35 is only 1 less than 36), we can state that √35 is slightly less than 6. Verification with a calculator gives us the answer 5.92 - we were right.

4. Another way is to decompose the root number into prime factors. Prime factors are numbers that are only divisible by 1 and themselves. Write the prime factors in a row and find pairs of identical factors. Such factors can be taken out of the sign of the root.

   • For example, calculate the square root of 45. We decompose the root number into prime factors: 45 \u003d 9 x 5, and 9 \u003d 3 x 3. Thus, √45 \u003d √ (3 x 3 x 5). 3 can be taken out of the root sign: √45 = 3√5. Now we can estimate √5.
   • Consider another example: √88.
     • = √(2 x 44)
     • = √ (2 x 4 x 11)
     • = √ (2 x 2 x 2 x 11). You got three multiplier 2s; take a couple of them and take them out of the sign of the root.
     • = 2√(2 x 11) = 2√2 x √11. Now we can evaluate √2 and √11 and find an approximate answer.

   Calculating the square root manually

   Using column division

   1. This method involves a process similar to long division and gives an accurate answer. First, draw a vertical line dividing the sheet into two halves, and then draw a horizontal line to the right and slightly below the top edge of the sheet to the vertical line. Now divide the root number into pairs of numbers, starting with the fractional part after the decimal point. So, the number 79520789182.47897 is written as "7 95 20 78 91 82, 47 89 70".

      • For example, let's calculate the square root of the number 780.14. Draw two lines (as shown in the picture) and write the number in the top left as "7 80, 14". It is normal that the first digit from the left is an unpaired digit. The answer (the root of the given number) will be written on the top right.

   2. Given the first pair of numbers (or one number) from the left, find the largest integer n whose square is less than or equal to the pair of numbers (or one number) in question. In other words, find the square number that is closest to, but less than, the first pair of numbers (or single number) from the left, and take the square root of that square number; you will get the number n. Write the found n at the top right, and write down the square n at the bottom right.

      • In our case, the first number on the left will be the number 7. Next, 4< 7, то есть 2 2 < 7 и n = 2. Напишите 2 сверху справа - это первая цифра в искомом квадратном корне. Напишите 2×2=4 справа снизу; вам понадобится это число для последующих вычислений.

   3. Subtract the square of the number n you just found from the first pair of numbers (or one number) from the left. Write the result of the calculation under the subtrahend (the square of the number n).

      • In our example, subtract 4 from 7 to get 3.

   4. Take down the second pair of numbers and write it down next to the value obtained in the previous step. Then double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the second pair of numbers is "80". Write "80" after the 3. Then, doubling the number from the top right gives 4. Write "4_×_=" from the bottom right.

   5. Fill in the blanks on the right.

      • In our case, if we put the number 8 instead of dashes, then 48 x 8 \u003d 384, which is more than 380. Therefore, 8 is too large a number, but 7 is fine. Write 7 instead of dashes and get: 47 x 7 \u003d 329. Write 7 from the top right - this is the second digit in the desired square root of the number 780.14.

   6. Subtract the resulting number from the current number on the left. Write the result from the previous step below the current number on the left, find the difference and write it below the subtracted one.

      • In our example, subtract 329 from 380, which equals 51.

   7. Repeat step 4. If the demolished pair of numbers is the fractional part of the original number, then put the separator (comma) of the integer and fractional parts in the desired square root from the top right. On the left, carry down the next pair of numbers. Double the number at the top right and write the result at the bottom right with "_×_=" appended.

      • In our example, the next pair of numbers to be demolished will be the fractional part of the number 780.14, so put the separator of the integer and fractional parts in the desired square root from the top right. Demolish 14 and write down at the bottom left. Double the top right (27) is 54, so write "54_×_=" at the bottom right.

   8. Repeat steps 5 and 6. Find the largest number in place of dashes on the right (instead of dashes you need to substitute the same number) so that the multiplication result is less than or equal to the current number on the left.

      • In our example, 549 x 9 = 4941, which is less than the current number on the left (5114). Write 9 on the top right and subtract the result of the multiplication from the current number on the left: 5114 - 4941 = 173.

   9. If you need to find more decimal places for the square root, write a pair of zeros next to the current number on the left and repeat steps 4, 5 and 6. Repeat steps until you get the accuracy of the answer you need (number of decimal places).

      Understanding the process

      1. To master this method, imagine the number whose square root you need to find as the area of ​​​​the square S. In this case, you will look for the length of the side L of such a square. Calculate the value of L for which L² = S.

         Enter a letter for each digit in your answer. Denote by A the first digit in the value of L (the desired square root). B will be the second digit, C the third and so on.

         Specify a letter for each pair of leading digits. Denote by S a the first pair of digits in the value S, by S b the second pair of digits, and so on.

         Explain the connection of this method with long division. As in the division operation, where each time we are only interested in one next digit of the divisible number, when calculating the square root, we work with a pair of digits in sequence (to obtain the next one digit in the square root value).

      2. Consider the first pair of digits Sa of the number S (Sa = 7 in our example) and find its square root. In this case, the first digit A of the sought value of the square root will be such a digit, the square of which is less than or equal to S a (that is, we are looking for such an A that satisfies the inequality A² ≤ Sa< (A+1)²). В нашем примере, S1 = 7, и 2² ≤ 7 < 3²; таким образом A = 2.

         • Let's say we need to divide 88962 by 7; here the first step will be similar: we consider the first digit of the divisible number 88962 (8) and select the largest number that, when multiplied by 7, gives a value less than or equal to 8. That is, we are looking for a number d for which the inequality is true: 7 × d ≤ 8< 7×(d+1). В этом случае d будет равно 1.

      3. Mentally imagine the square whose area you need to calculate. You are looking for L, that is, the length of the side of a square whose area is S. A, B, C are numbers in the number L. You can write it differently: 10A + B \u003d L (for a two-digit number) or 100A + 10B + C \u003d L (for three-digit number) and so on.

         • Let be (10A+B)² = L² = S = 100A² + 2×10A×B + B². Remember that 10A+B is a number whose B stands for ones and A stands for tens. For example, if A=1 and B=2, then 10A+B equals the number 12. (10A+B)² is the area of ​​the whole square, 100A² is the area of ​​the large inner square, B² is the area of ​​the small inner square, 10A×B is the area of ​​each of the two rectangles. Adding the areas of the figures described, you will find the area of ​​the original square.

Root formulas. properties of square roots.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

In the previous lesson, we figured out what a square root is. It's time to figure out what are formulas for roots, what are root properties and what can be done about it all.

Root Formulas, Root Properties, and Rules for Actions with Roots- it's essentially the same thing. There are surprisingly few formulas for square roots. Which, of course, pleases! Rather, you can write a lot of all sorts of formulas, but only three are enough for practical and confident work with roots. Everything else flows from these three. Although many stray in the three formulas of the roots, yes ...

Let's start with the simplest. Here she is:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Among the many knowledge that is a sign of literacy, the alphabet is in the first place. The next, the same "sign" element, are the skills of addition-multiplication and, adjacent to them, but reverse in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, And everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to take roots in life, except in the country? For example, such an entertaining problem, like, the square root of the number 12345 ... Is there still gunpowder in the powder flasks? Can we do it? Yes, there is nothing easier! Where is my calculator ... And without it, hand-to-hand, weak?

First, let's clarify what it is - the square root of a number. Generally speaking, "to extract the root from a number" means to perform the arithmetic operation opposite to raising to a power - here you have the unity of opposites in life application. let's say a square is a multiplication of a number by itself, i.e., as they taught at school, X * X = A or in another notation X2 = A, and in words - “X squared equals A”. Then the inverse problem sounds like this: the square root of the number A, is the number X, which, when squared, is equal to A.

Extracting the square root

From the school course of arithmetic, methods of calculating "in a column" are known, which help to perform any calculations using the first four arithmetic operations. Alas ... For square, and not only square, roots of such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequential enumeration of numbers, the square of which approaches the value of the root expression. Only and everything! An hour or two will not have time to pass, as you can calculate using the well-known method of multiplying into a "column", any square root. If you have the skills, a couple of minutes is enough for this. Even a not quite advanced calculator or PC user does it in one fell swoop - progress.

But seriously, the calculation of the square root is often performed using the “artillery fork” technique: first, they take a number whose square approximately corresponds to the root expression. It is better if "our square" is slightly less than this expression. Then they correct the number according to their own skill-understanding, for example, multiply by two, and ... square it again. If the result is greater than the number under the root, successively adjusting the original number, gradually approaching its "colleague" under the root. As you can see - no calculator, only the ability to count "in a column". Of course, there are many scientifically reasoned and optimized algorithms for calculating the square root, but for "home use" the above technique gives 100% confidence in the result.

Yes, I almost forgot, in order to confirm our increased literacy, we calculate the square root of the previously indicated number 12345. We do it step by step:

1. Take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is on top - the result is less than 12345.

2. Let's try, also purely intuitively, X = 120. Then: X * X = 14400. And again, with intuition, the order - the result is more than 12345.

3. Above, a “fork” of 100 and 120 is obtained. Let's choose new numbers - 110 and 115. We get, respectively, 12100 and 13225 - the fork narrows.

4. We try on "maybe" X = 111. We get X * X = 12321. This number is already quite close to 12345. In accordance with the required accuracy, the “fitting” can be continued or stopped at the result obtained. That's all. As promised - everything is very simple and without a calculator.

Quite a bit of history...

Even the Pythagoreans, students of the school and followers of Pythagoras, thought of using square roots, 800 BC. and right there, "ran into" new discoveries in the field of numbers. And where did it come from?

1. The solution of the problem with the extraction of the root, gives the result in the form of numbers of a new class. They were called irrational, in other words, "unreasonable", because. they are not written as a complete number. The most classic example of this kind is the square root of 2. This case corresponds to the calculation of the diagonal of a square with a side equal to 1 - here it is, the influence of the Pythagorean school. It turned out that in a triangle with a very specific unit size of the sides, the hypotenuse has a size that is expressed by a number that "has no end." So in mathematics appeared

2. It is known that It turned out that this mathematical operation contains one more catch - extracting the root, we do not know what square of which number, positive or negative, is the root expression. This uncertainty, the double result from one operation, is written down.

The study of the problems associated with this phenomenon has become a direction in mathematics called the theory of a complex variable, which is of great practical importance in mathematical physics.

It is curious that the designation of the root - radical - was used in his "Universal Arithmetic" by the same ubiquitous I. Newton, and exactly the modern form of writing the root has been known since 1690 from the Frenchman Roll's book "Algebra Manual".

In this article, we will introduce the concept of the root of a number. We will act sequentially: we will start with the square root, from it we will move on to the description of the cube root, after that we will generalize the concept of the root by defining the root of the nth degree. At the same time, we will introduce definitions, notation, give examples of roots and give the necessary explanations and comments.

Square root, arithmetic square root

To understand the definition of the root of a number, and the square root in particular, one must have . At this point, we will often encounter the second power of a number - the square of a number.

Let's start with square root definitions.

Definition

The square root of a is the number whose square is a .

In order to bring examples of square roots, take several numbers, for example, 5 , −0.3 , 0.3 , 0 , and square them, we get the numbers 25 , 0.09 , 0.09 and 0 respectively (5 2 \u003d 5 5 \u003d 25 , (−0.3) 2 =(−0.3) (−0.3)=0.09, (0.3) 2 =0.3 0.3=0.09 and 0 2 =0 0=0 ). Then by the definition above, 5 is the square root of 25, −0.3 and 0.3 are the square roots of 0.09, and 0 is the square root of zero.

It should be noted that not for any number a exists , whose square is equal to a . Namely, for any negative number a, there is no real number b whose square is equal to a. Indeed, the equality a=b 2 is impossible for any negative a , since b 2 is a non-negative number for any b . Thus, on the set of real numbers there is no square root of a negative number. In other words, on the set of real numbers, the square root of a negative number is not defined and has no meaning.

This leads to a logical question: “Is there a square root of a for any non-negative a”? The answer is yes. The rationale for this fact can be considered a constructive method used to find the value of the square root.

Then the following logical question arises: "What is the number of all square roots of a given non-negative number a - one, two, three, or even more"? Here is the answer to it: if a is zero, then the only square root of zero is zero; if a is some positive number, then the number of square roots from the number a is equal to two, and the roots are . Let's substantiate this.

Let's start with the case a=0 . Let us first show that zero is indeed the square root of zero. This follows from the obvious equality 0 2 =0·0=0 and the definition of the square root.

Now let's prove that 0 is the only square root of zero. Let's use the opposite method. Let's assume that there is some non-zero number b that is the square root of zero. Then the condition b 2 =0 must be satisfied, which is impossible, since for any non-zero b the value of the expression b 2 is positive. We have come to a contradiction. This proves that 0 is the only square root of zero.

Let's move on to cases where a is a positive number. Above we said that there is always a square root of any non-negative number, let b be the square root of a. Let's say that there is a number c , which is also the square root of a . Then, by the definition of the square root, the equalities b 2 =a and c 2 =a are valid, from which it follows that b 2 −c 2 =a−a=0, but since b 2 −c 2 =(b−c) ( b+c) , then (b−c) (b+c)=0 . The resulting equality in force properties of actions with real numbers only possible when b−c=0 or b+c=0 . Thus the numbers b and c are equal or opposite.

If we assume that there is a number d, which is another square root of the number a, then by reasoning similar to those already given, it is proved that d is equal to the number b or the number c. So, the number of square roots of a positive number is two, and the square roots are opposite numbers.

For the convenience of working with square roots, the negative root is "separated" from the positive one. For this purpose, it introduces definition of arithmetic square root.

Definition

Arithmetic square root of a non-negative number a is a non-negative number whose square is equal to a .

For the arithmetic square root of the number a, the notation is accepted. The sign is called the arithmetic square root sign. It is also called the sign of the radical. Therefore, you can partly hear both "root" and "radical", which means the same object.

The number under the arithmetic square root sign is called root number, and the expression under the root sign - radical expression, while the term "radical number" is often replaced by "radical expression". For example, in the notation, the number 151 is a radical number, and in the notation, the expression a is a radical expression.

When reading, the word "arithmetic" is often omitted, for example, the entry is read as "the square root of seven point twenty-nine hundredths." The word "arithmetic" is pronounced only when they want to emphasize that we are talking about the positive square root of a number.

In the light of the introduced notation, it follows from the definition of the arithmetic square root that for any non-negative number a .

The square roots of a positive number a are written using the arithmetic square root sign as and . For example, the square roots of 13 are and . The arithmetic square root of zero is zero, that is, . For negative numbers a, we will not attach meaning to the entries until we study complex numbers. For example, the expressions and are meaningless.

Based on the definition of a square root, properties of square roots are proved, which are often used in practice.

To conclude this subsection, we note that the square roots of a number are solutions of the form x 2 =a with respect to the variable x .

cube root of

Definition of the cube root of the number a is given in a similar way to the definition of the square root. Only it is based on the concept of a cube of a number, not a square.

Definition

The cube root of a a number whose cube is equal to a is called.

Let's bring examples of cube roots. To do this, take several numbers, for example, 7 , 0 , −2/3 , and cube them: 7 3 =7 7 7=343 , 0 3 =0 0 0=0 , . Then, based on the definition of the cube root, we can say that the number 7 is the cube root of 343, 0 is the cube root of zero, and −2/3 is the cube root of −8/27.

It can be shown that the cube root of the number a, unlike the square root, always exists, and not only for non-negative a, but also for any real number a. To do this, you can use the same method that we mentioned when studying the square root.

Moreover, there is only one cube root of a given number a. Let us prove the last assertion. To do this, consider three cases separately: a is a positive number, a=0 and a is a negative number.

It is easy to show that for positive a, the cube root of a cannot be either negative or zero. Indeed, let b be the cube root of a , then by definition we can write the equality b 3 =a . It is clear that this equality cannot be true for negative b and for b=0, since in these cases b 3 =b·b·b will be a negative number or zero, respectively. So the cube root of a positive number a is a positive number.

Now suppose that in addition to the number b there is one more cube root from the number a, let's denote it c. Then c 3 =a. Therefore, b 3 −c 3 =a−a=0 , but b 3 −c 3 =(b−c) (b 2 +b c+c 2)(this is the abbreviated multiplication formula difference of cubes), whence (b−c) (b 2 +b c+c 2)=0 . The resulting equality is only possible when b−c=0 or b 2 +b c+c 2 =0 . From the first equality we have b=c , and the second equality has no solutions, since its left side is a positive number for any positive numbers b and c as the sum of three positive terms b 2 , b c and c 2 . This proves the uniqueness of the cube root of a positive number a.

For a=0, the only cube root of a is zero. Indeed, if we assume that there is a number b , which is a non-zero cube root of zero, then the equality b 3 =0 must hold, which is possible only when b=0 .

For negative a , one can argue similar to the case for positive a . First, we show that the cube root of a negative number cannot be equal to either a positive number or zero. Secondly, we assume that there is a second cube root of a negative number and show that it will necessarily coincide with the first one.

So, there is always a cube root of any given real number a, and only one.

Let's give definition of arithmetic cube root.

Definition

Arithmetic cube root of a non-negative number a a non-negative number whose cube is equal to a is called.

The arithmetic cube root of a non-negative number a is denoted as , the sign is called the sign of the arithmetic cube root, the number 3 in this notation is called root indicator. The number under the root sign is root number, the expression under the root sign is radical expression.

Although the arithmetic cube root is defined only for non-negative numbers a, it is also convenient to use entries in which negative numbers are under the arithmetic cube root sign. We will understand them as follows: , where a is a positive number. For example, .

We will talk about the properties of cube roots in the general article properties of roots.

Calculating the value of a cube root is called extracting a cube root, this action is discussed in the article extracting roots: methods, examples, solutions.

To conclude this subsection, we say that the cube root of a is a solution of the form x 3 =a.

Nth root, arithmetic root of n

We generalize the concept of a root from a number - we introduce determination of the nth root for n.

Definition

nth root of a is a number whose nth power is equal to a.

From this definition it is clear that the root of the first degree from the number a is the number a itself, since when studying the degree with a natural indicator, we took a 1 = a.

Above, we considered special cases of the root of the nth degree for n=2 and n=3 - the square root and the cube root. That is, the square root is the root of the second degree, and the cube root is the root of the third degree. To study the roots of the nth degree for n=4, 5, 6, ..., it is convenient to divide them into two groups: the first group - the roots of even degrees (that is, for n=4, 6, 8, ...), the second group - the roots odd powers (that is, for n=5, 7, 9, ... ). This is due to the fact that the roots of even degrees are similar to the square root, and the roots of odd degrees are similar to the cubic root. Let's deal with them in turn.

Let's start with the roots, the powers of which are the even numbers 4, 6, 8, ... As we have already said, they are similar to the square root of the number a. That is, the root of any even degree from the number a exists only for non-negative a. Moreover, if a=0, then the root of a is unique and equal to zero, and if a>0, then there are two roots of an even degree from the number a, and they are opposite numbers.

Let us justify the last assertion. Let b be a root of an even degree (we denote it as 2·m, where m is some natural number) from a. Suppose there is a number c - another 2 m root of a . Then b 2 m −c 2 m =a−a=0 . But we know of the form b 2 m − c 2 m = (b − c) (b + c) (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2), then (b−c) (b+c) (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2)=0. From this equality it follows that b−c=0 , or b+c=0 , or b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2 =0. The first two equalities mean that the numbers b and c are equal or b and c are opposite. And the last equality is valid only for b=c=0 , since its left side contains an expression that is non-negative for any b and c as the sum of non-negative numbers.

As for the roots of the nth degree for odd n, they are similar to the cube root. That is, the root of any odd degree from the number a exists for any real number a, and for a given number a it is unique.

The uniqueness of the root of odd degree 2·m+1 from the number a is proved by analogy with the proof of the uniqueness of the cube root from a . Only here instead of equality a 3 −b 3 =(a−b) (a 2 +a b+c 2) an equality of the form b 2 m+1 −c 2 m+1 = (b−c) (b 2 m +b 2 m−1 c+b 2 m−2 c 2 +… +c 2 m). The expression in the last parenthesis can be rewritten as b 2 m +c 2 m +b c (b 2 m−2 +c 2 m−2 + b c (b 2 m−4 +c 2 m−4 +b c (…+(b 2 +c 2 +b c)))). For example, for m=2 we have b 5 −c 5 =(b−c) (b 4 +b 3 c+b 2 c 2 +b c 3 +c 4)= (b−c) (b 4 +c 4 +b c (b 2 +c 2 +b c)). When a and b are both positive or both negative, their product is a positive number, then the expression b 2 +c 2 +b·c , which is in the parentheses of the highest degree of nesting, is positive as the sum of positive numbers. Now, moving successively to the expressions in brackets of the previous degrees of nesting, we make sure that they are also positive as the sums of positive numbers. As a result, we obtain that the equality b 2 m+1 −c 2 m+1 = (b−c) (b 2 m +b 2 m−1 c+b 2 m−2 c 2 +… +c 2 m)=0 only possible when b−c=0 , that is, when the number b is equal to the number c .

It's time to deal with the notation of the roots of the nth degree. For this, it is given determination of the arithmetic root of the nth degree.

Definition

The arithmetic root of the nth degree of a non-negative number a a non-negative number is called, the nth power of which is equal to a.

The area of ​​a square plot of land is 81 dm². Find his side. Suppose the length of the side of the square is X decimetres. Then the area of ​​the plot is X² square decimetres. Since, according to the condition, this area is 81 dm², then X² = 81. The length of the side of a square is a positive number. A positive number whose square is 81 is the number 9. When solving the problem, it was required to find the number x, the square of which is 81, i.e. solve the equation X² = 81. This equation has two roots: x 1 = 9 and x 2 \u003d - 9, since 9² \u003d 81 and (- 9)² \u003d 81. Both numbers 9 and - 9 are called the square roots of the number 81.

Note that one of the square roots X= 9 is a positive number. It is called the arithmetic square root of 81 and is denoted √81, so √81 = 9.

Arithmetic square root of a number a is a non-negative number whose square is equal to a.

For example, the numbers 6 and -6 are the square roots of 36. The number 6 is the arithmetic square root of 36, since 6 is a non-negative number and 6² = 36. The number -6 is not an arithmetic root.

Arithmetic square root of a number a denoted as follows: √ a.

The sign is called the arithmetic square root sign; a is called a root expression. Expression √ a read like this: the arithmetic square root of a number a. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In cases where it is clear that we are talking about an arithmetic root, they briefly say: "the square root of a«.

The act of finding the square root of a number is called taking the square root. This action is the reverse of squaring.

Any number can be squared, but not every number can be square roots. For example, it is impossible to extract the square root of the number - 4. If such a root existed, then, denoting it with the letter X, we would get the wrong equality x² \u003d - 4, since there is a non-negative number on the left, and a negative one on the right.

Expression √ a only makes sense when a ≥ 0. The definition of the square root can be briefly written as: √ a ≥ 0, (√a)² = a. Equality (√ a)² = a valid for a ≥ 0. Thus, to make sure that the square root of a non-negative number a equals b, i.e., that √ a =b, you need to check that the following two conditions are met: b ≥ 0, b² = a.

The square root of a fraction

Let's calculate . Note that √25 = 5, √36 = 6, and check if the equality holds.

As and , then the equality is true. So, .

Theorem: If a a≥ 0 and b> 0, that is, the root of the fraction is equal to the root of the numerator divided by the root of the denominator. It is required to prove that: and .

Since √ a≥0 and √ b> 0, then .

By the property of raising a fraction to a power and determining the square root the theorem is proven. Let's look at a few examples.

Calculate , according to the proven theorem .

Second example: Prove that , if a ≤ 0, b < 0. .

Another example: Calculate .

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Square root transformation

Taking the multiplier out from under the sign of the root. Let an expression be given. If a a≥ 0 and b≥ 0, then by the theorem on the root of the product, we can write:

Such a transformation is called factoring out the root sign. Consider an example;

Calculate at X= 2. Direct substitution X= 2 in the radical expression leads to complicated calculations. These calculations can be simplified if we first remove the factors from under the root sign: . Now substituting x = 2, we get:.

So, when taking out the factor from under the root sign, the radical expression is represented as a product in which one or more factors are the squares of non-negative numbers. The root product theorem is then applied and the root of each factor is taken. Consider an example: Simplify the expression A = √8 + √18 - 4√2 by taking out the factors from under the root sign in the first two terms, we get:. We emphasize that the equality valid only when a≥ 0 and b≥ 0. if a < 0, то .