How to solve the square root. How to quickly extract square roots

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 - Square root numbers. 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 calculations "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...

Thinking about using square roots still the Pythagoreans, students of the school and followers of Pythagoras, for 800 years 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 root designation - radical - was used in his "Universal Arithmetic" by the same ubiquitous I. Newton, but exactly modern look The root record has been known since 1690 from the book of the Frenchman Roll "Guide to Algebra".

Mathematics was born when a person became aware of himself and began to position himself as an autonomous unit of the world. The desire to measure, compare, calculate what surrounds you is what underlay one of the fundamental sciences of our days. At first, these were particles of elementary mathematics, which made it possible to connect numbers with their physical expressions, later the conclusions began to be presented only theoretically (because of their abstractness), but after a while, as one scientist put it, "mathematics reached the ceiling of complexity when all numbers." The concept of "square root" appeared at a time when it could be easily supported by empirical data, going beyond the plane of calculations.

How it all started

The first mention of the root, which on this moment denoted as √, was recorded in the writings of the Babylonian mathematicians, who laid the foundation for modern arithmetic. Of course, they looked a little like the current form - the scientists of those years first used bulky tablets. But in the second millennium BC. e. they came up with an approximate calculation formula that showed how to take the square root. The photo below shows a stone on which Babylonian scientists carved the output process √2, and it turned out to be so correct that the discrepancy in the answer was found only in the tenth decimal place.

In addition, the root was used if it was necessary to find the side of a triangle, provided that the other two were known. Well, when solving quadratic equations, there is no escape from extracting the root.

Along with the Babylonian works, the object of the article was also studied in the Chinese work "Mathematics in Nine Books", and the ancient Greeks came to the conclusion that any number from which the root is not extracted without a remainder gives an irrational result.

The origin of this term is associated with the Arabic representation of the number: ancient scientists believed that the square of an arbitrary number grows from the root, like a plant. In Latin, this word sounds like radix (one can trace a pattern - everything that has a "root" semantic load is consonant, be it radish or sciatica).

Scientists of subsequent generations picked up this idea, designating it as Rx. For example, in the 15th century, in order to indicate that the square root is taken from an arbitrary number a, they wrote R 2 a. Habitual modern look"tick" √ appeared only in the 17th century thanks to Rene Descartes.

Our days

Mathematically, the square root of y is the number z whose square is y. In other words, z 2 =y is equivalent to √y=z. However, this definition is relevant only for the arithmetic root, since it implies a non-negative value of the expression. In other words, √y=z, where z is greater than or equal to 0.

In general, which is valid for determining an algebraic root, the value of an expression can be either positive or negative. Thus, due to the fact that z 2 =y and (-z) 2 =y, we have: √y=±z or √y=|z|.

Due to the fact that love for mathematics has only increased with the development of science, there are various manifestations of affection for it, not expressed in dry calculations. For example, along with such interesting events as the day of Pi, the holidays of the square root are also celebrated. They are celebrated nine times in a hundred years, and are determined according to the following principle: the numbers that denote the day and month in order must be the square root of the year. Yes, in next time This holiday will be celebrated on April 4, 2016.

Properties of the square root on the field R

Almost all mathematical expressions have a geometric basis, this fate did not pass and √y, which is defined as the side of a square with area y.

How to find the root of a number?

There are several calculation algorithms. The simplest, but at the same time quite cumbersome, is the usual arithmetic calculation, which is as follows:

1) from the number whose root we need, odd numbers are subtracted in turn - until the remainder at the output is less than the subtracted one or even zero. The number of moves will eventually become the desired number. For example, calculating the square root of 25:

The next odd number is 11, the remainder is: 1<11. Количество ходов - 5, так что корень из 25 равен 5. Вроде все легко и просто, но представьте, что придется вычислять из 18769?

For such cases, there is a Taylor series expansion:

√(1+y)=∑((-1) n (2n)!/(1-2n)(n!) 2 (4 n))y n , where n takes values ​​from 0 to

+∞, and |y|≤1.

Graphic representation of the function z=√y

Consider an elementary function z=√y on the field of real numbers R, where y is greater than or equal to zero. Her chart looks like this:

The curve grows from the origin and necessarily crosses the point (1; 1).

Properties of the function z=√y on the field of real numbers R

1. The domain of definition of the considered function is the interval from zero to plus infinity (zero is included).

2. The range of values ​​of the considered function is the interval from zero to plus infinity (zero is again included).

3. The function takes the minimum value (0) only at the point (0; 0). There is no maximum value.

4. The function z=√y is neither even nor odd.

5. The function z=√y is not periodic.

6. There is only one point of intersection of the graph of the function z=√y with the coordinate axes: (0; 0).

7. The intersection point of the graph of the function z=√y is also the zero of this function.

8. The function z=√y is continuously growing.

9. The function z=√y takes only positive values, therefore, its graph occupies the first coordinate angle.

Options for displaying the function z=√y

In mathematics, to facilitate the calculation of complex expressions, the power form of writing the square root is sometimes used: √y=y 1/2. This option is convenient, for example, in raising a function to a power: (√y) 4 =(y 1/2) 4 =y 2 . This method is also a good representation for differentiation with integration, since thanks to it the square root is represented by an ordinary power function.

And in programming, the replacement for the symbol √ is the combination of letters sqrt.

It is worth noting that in this area the square root is in great demand, as it is part of most of the geometric formulas necessary for calculations. The counting algorithm itself is quite complicated and is based on recursion (a function that calls itself).

The square root in the complex field C

By and large, it was the subject of this article that stimulated the discovery of the field of complex numbers C, since mathematicians were haunted by the question of obtaining an even degree root from a negative number. This is how the imaginary unit i appeared, which is characterized by a very interesting property: its square is -1. Thanks to this, quadratic equations and with a negative discriminant got a solution. In C, for the square root, the same properties are relevant as in R, the only thing is that the restrictions on the root expression are removed.

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 the 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 equal to the root from 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 .

.

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 .

Quite often, when solving problems, we are faced with large numbers from which we need to extract Square root. Many students decide that this is a mistake and start resolving the whole example. Under no circumstances should this be done! There are two reasons for this:

1. Roots from big numbers actually occur in tasks. Especially in text;
2. There is an algorithm by which these roots are considered almost verbally.

We will consider this algorithm today. Perhaps some things will seem incomprehensible to you. But if you pay attention to this lesson, you will get the most powerful weapon against square roots.

So the algorithm:

1. Restrict the desired root above and below to multiples of 10. Thus, we will reduce the search range to 10 numbers;
2. From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
3. Square these 1-2 numbers. That of them, the square of which is equal to the original number, will be the root.

Before applying this algorithm works in practice, let's look at each individual step.

Roots constraint

First of all, we need to find out between which numbers our root is located. It is highly desirable that the numbers be a multiple of ten:

10 2 = 100;
20 2 = 400;
30 2 = 900;
40 2 = 1600;
...
90 2 = 8100;
100 2 = 10 000.

We get a series of numbers:

100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10 000.

What do these numbers give us? It's simple: we get boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and greater than 40:

[Figure caption]

The same is with any other number from which you can find the square root. For example, 3364:

[Figure caption]

Thus, instead of an incomprehensible number, we get a very specific range in which the original root lies. To further narrow the scope of the search, go to the second step.

Elimination of obviously superfluous numbers

So, we have 10 numbers - candidates for the root. We received them very quickly, without complex thinking and multiplication in a column. It's time to move on.

Believe it or not, now we will reduce the number of candidate numbers to two - and again without any complicated calculations! enough to know special rule. Here it is:

The last digit of the square depends only on the last digit original number.

In other words, it is enough to look at the last digit of the square - and we will immediately understand where the original number ends.

There are only 10 digits that can stand on last place. Let's try to find out what they turn into when they are squared. Take a look at the table:

1	2	3	4	5	6	7	8	9	0
1	4	9	6	5	6	9	4	1	0

This table is another step towards calculating the root. As you can see, the numbers in the second line turned out to be symmetrical with respect to the five. For example:

2 2 = 4;
8 2 = 64 → 4.

As you can see, the last digit is the same in both cases. And this means that, for example, the root of 3364 necessarily ends in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

[Figure caption]

The red squares show that we don't know this figure yet. But after all, the root lies between 50 and 60, on which there are only two numbers ending in 2 and 8:

[Figure caption]

That's all! Of all the possible roots, we left only two options! And this is in the most difficult case, because the last digit can be 5 or 0. And then the only candidate for the roots will remain!

Final Calculations

So, we have 2 candidate numbers left. How do you know which one is the root? The answer is obvious: square both numbers. The one that squared will give the original number, and will be the root.

For example, for the number 3364, we found two candidate numbers: 52 and 58. Let's square them:

52 2 \u003d (50 +2) 2 \u003d 2500 + 2 50 2 + 4 \u003d 2704;
58 2 \u003d (60 - 2) 2 \u003d 3600 - 2 60 2 + 4 \u003d 3364.

That's all! It turned out that the root is 58! At the same time, in order to simplify the calculations, I used the formula of the squares of the sum and difference. Thanks to this, you didn’t even have to multiply the numbers in a column! This is another level of optimization of calculations, but, of course, it is completely optional :)

Root Calculation Examples

Theory is good, of course. But let's test it in practice.

[Figure caption]

First, let's find out between which numbers the number 576 lies:

400 < 576 < 900
20 2 < 576 < 30 2

Now let's look at the last number. It is equal to 6. When does this happen? Only if the root ends in 4 or 6. We get two numbers:

It remains to square each number and compare with the original:

24 2 = (20 + 4) 2 = 576

Fine! The first square turned out to be equal to the original number. So this is the root.

Task. Calculate the square root:

[Figure caption]

900 < 1369 < 1600;
30 2 < 1369 < 40 2;

Let's look at the last number:

1369 → 9;
33; 37.

Let's square it:

33 2 \u003d (30 + 3) 2 \u003d 900 + 2 30 3 + 9 \u003d 1089 ≠ 1369;
37 2 \u003d (40 - 3) 2 \u003d 1600 - 2 40 3 + 9 \u003d 1369.

Here is the answer: 37.

Task. Calculate the square root:

[Figure caption]

We limit the number:

2500 < 2704 < 3600;
50 2 < 2704 < 60 2;

Let's look at the last number:

2704 → 4;
52; 58.

Let's square it:

52 2 = (50 + 2) 2 = 2500 + 2 50 2 + 4 = 2704;

We got the answer: 52. The second number will no longer need to be squared.

Task. Calculate the square root:

[Figure caption]

We limit the number:

3600 < 4225 < 4900;
60 2 < 4225 < 70 2;

Let's look at the last number:

4225 → 5;
65.

As you can see, after the second step, only one option remains: 65. This is the desired root. But let's still square it and check:

65 2 = (60 + 5) 2 = 3600 + 2 60 5 + 25 = 4225;

Everything is correct. We write down the answer.

Conclusion

Alas, no better. Let's take a look at the reasons. There are two of them:

• It is forbidden to use calculators at any normal math exam, be it the GIA or the Unified State Examination. And for carrying a calculator into the classroom, they can easily be kicked out of the exam.
• Don't be like stupid Americans. Which are not like roots - they cannot add two prime numbers. And at the sight of fractions, they generally get hysterical.

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 cube root, after that we 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 are no real number b , whose square would be 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 negative root separates from the positive. 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.

In conclusion of this subsection, we note that the square roots of the number a 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 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.

We start with roots whose powers are even numbers 4, 6, 8, ... As we have already said, they are analogous to the square root of 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 even degree (we denote it as 2 m, where m is some natural number) from number 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.