Root degree n from a real number a, Where n- natural number, such a real number is called x, n the th power of which is equal to a.

Root degree n from among a is indicated by the symbol. According to this definition.

Finding the root n-th degree from among a called root extraction. Number A is called a radical number (expression), n- root indicator. For odd n there is a root n-th power for any real number a. When even n there is a root n-th power only for non-negative numbers a. To disambiguate the root n-th degree from among a, the concept of an arithmetic root is introduced n-th degree from among a.

The concept of an arithmetic root of degree N

If and n- natural number, greater 1 , then there is, and only one, non-negative number X, such that the equality is satisfied. This number X called an arithmetic root n th power of a non-negative number A and is designated . Number A is called a radical number, n- root indicator.

So, according to the definition, the notation , where , means, firstly, that and, secondly, that, i.e. .

The concept of a degree with a rational exponent

Degree with natural exponent: let A is a real number, and n- a natural number greater than one, n-th power of the number A call the work n factors, each of which is equal A, i.e. . Number A- the basis of the degree, n- exponent. A power with a zero exponent: by definition, if , then . Zero power of a number 0 doesn't make sense. A degree with a negative integer exponent: assumed by definition if and n is a natural number, then . A degree with a fractional exponent: it is assumed by definition if and n- natural number, m is an integer, then .

Operations with roots.

In all the formulas below, the symbol means an arithmetic root (the radical expression is positive).

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 roots of the dividend and the divisor:

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 n times and at the same time raise the radical number to the nth power, then the value of the root will not change:

5. If you reduce the degree of the root by n times and simultaneously extract the nth root of the radical number, then the value of the root will not change:

Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero and fractional exponents. All these exponents require additional definition.

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

Now the formula a m: a n = a m - n can be used not only for m greater than n, but also for m less than n.

EXAMPLE a 4: a 7 = a 4 - 7 = a -3.

If we want the formula a m: a n = a m - n to be valid for m = n, we need a definition of degree zero.

A degree with a zero index. The power of any non-zero number with exponent zero is 1.

EXAMPLES. 2 0 = 1, (– 5) 0 = 1, (– 3 / 5) 0 = 1.

Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:

About expressions that have no meaning. There are several such expressions.

Case 1.

Where a ≠ 0 does not exist.

In fact, if we assume that x is a certain number, then in accordance with the definition of the division operation we have: a = 0 x, i.e. a = 0, which contradicts the condition: a ≠ 0

Case 2.

Any number.

In fact, if we assume that this expression is equal to a certain number x, then according to the definition of the division operation we have: 0 = 0 x. But this equality holds for any number x, which is what needed to be proven.

Really,

Solution. Let's consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) for x > 0 we get: x / x = 1, i.e. 1 = 1, which means that x is any number; but taking into account that in our case x > 0, the answer is x > 0;

3) at x< 0 получаем: – x / x = 1, т.e. –1 = 1, следовательно,

in this case there is no solution. Thus x > 0.

An arithmetic root of the nth power of a non-negative number is a non-negative number whose nth power is equal to:

The power of a root is a natural number greater than 1.

3.

4.

Special cases:

1. If the root exponent is an odd integer(), then the radical expression can be negative.

In the case of an odd exponent, the equation for any real value and integer ALWAYS has a single root:

For a root of odd degree the following identity holds:

,

2. If the root exponent is an even integer (), then the radical expression cannot be negative.

In the case of an even exponent, Eq. has

at single root

and, if and

For a root of even degree the following identity holds:

For a root of even degree the following equalities are valid::

Power function, its properties and graph.

Power function and its properties.

Power function with natural exponent. The function y = x n, where n is a natural number, is called a power function with a natural exponent. For n = 1 we obtain the function y = x, its properties:

Direct proportionality. Direct proportionality is a function defined by the formula y = kx n, where the number k is called the proportionality coefficient.

Let us list the properties of the function y = kx.

The domain of a function is the set of all real numbers.

y = kx - odd function (f(- x) = k (- x)= - kx = -k(x)).

3) For k > 0 the function increases, and for k< 0 убывает на всей числовой прямой.

The graph (straight line) is shown in Figure II.1.

Rice. II.1.

When n=2 we get the function y = x 2, its properties:

Function y -x 2. Let us list the properties of the function y = x 2.

y = x 2 - even function (f(- x) = (- x) 2 = x 2 = f (x)).

The function decreases over the interval.

In fact, if , then - x 1 > - x 2 > 0, and therefore

(-x 1) 2 > (- x 2) 2, i.e., and this means the function is decreasing.

The graph of the function y=x2 is a parabola. This graph is shown in Figure II.2.

Rice. II.2.

When n = 3 we get the function y = x 3, its properties:

The domain of definition of a function is the entire number line.

y = x 3 - odd function (f (- x) = (- x) 2 = - x 3 = - f (x)).

3) The function y = x 3 increases along the entire number line. The graph of the function y = x 3 is shown in the figure. It is called a cubic parabola.

The graph (cubic parabola) is shown in Figure II.3.

Rice. II.3.

Let n be an arbitrary even natural number greater than two:

n = 4, 6, 8,... . In this case, the function y = x n has the same properties as the function y = x 2. The graph of such a function resembles a parabola y = x 2, only the branches of the graph at |n| >1 the steeper they go upward, the larger n, and the more “pressed” to the x axis, the larger n.

Let n be an arbitrary odd number greater than three: n = = 5, 7, 9, ... . In this case, the function y = x n has the same properties as the function y = x 3. The graph of such a function resembles a cubic parabola (only the branches of the graph go up and down the steeper, the larger n is. Note also that on the interval (0; 1) the graph of the power function y = x n moves away from the x axis more slowly as x increases, the more more than n.

Power function with negative integer exponent. Consider the function y = x - n, where n is a natural number. When n = 1 we get y = x - n or y = Properties of this function:

The graph (hyperbola) is shown in Figure II.4.

Entry level

Root and its properties. Detailed theory with examples (2019)

Let's try to figure out what kind of concept this “root” is and “what it is eaten with.” To do this, let's look at examples that you have already encountered in class (well, or you are just about to encounter this).

For example, we have an equation. What is the solution to this equation? What numbers can be squared and obtained? Remembering the multiplication table, you can easily give the answer: and (after all, when two negative numbers are multiplied, a positive number is obtained)! To simplify, mathematicians introduced the special concept of the square root and assigned it a special symbol.

Let us define the arithmetic square root.

Why does the number have to be non-negative? For example, what is it equal to? Well, well, let's try to pick one. Maybe three? Let's check: , not. Maybe, ? Again, we check: . Well, it doesn’t fit? This is to be expected - because there are no numbers that, when squared, give a negative number!
This is what you need to remember: the number or expression under the root sign must be non-negative!

However, the most attentive ones have probably already noticed that the definition says that the solution to the square root of “a number is called this non-negative number whose square is equal to ". Some of you will say that at the very beginning we looked at an example, selected numbers that can be squared and get, the answer was and, but here we are talking about some kind of “non-negative number”! This remark is quite appropriate. Here you just need to distinguish between the concepts of quadratic equations and the arithmetic square root of a number. For example, is not equivalent to the expression.

It follows that, that is, or. (Read the topic "")

And it follows that.

Of course, this is very confusing, but it is necessary to remember that the signs are the result of solving the equation, since when solving the equation we must write down all the X's, which, when substituted into the original equation, will give the correct result. Both and fit into our quadratic equation.

However, if just take the square root from something, then always we get one non-negative result.

Now try to solve this equation. Everything is not so simple and smooth anymore, is it? Try going through the numbers, maybe something will work out? Let's start from the very beginning - from scratch: - doesn't fit, move on - less than three, also sweep aside, what if. Let's check: - also not suitable, because... that's more than three. It's the same story with negative numbers. So what should we do now? Did the search really give us nothing? Not at all, now we know for sure that the answer will be some number between and, as well as between and. Also, obviously the solutions won't be integers. Moreover, they are not rational. So what next? Let's graph the function and mark the solutions on it.

Let's try to cheat the system and get the answer using a calculator! Let's get the root out of it! Oh-oh-oh, it turns out that. This number never ends. How can you remember this, since there won’t be a calculator on the exam!? Everything is very simple, you don’t need to remember it, you just need to remember (or be able to quickly estimate) the approximate value. and the answers themselves. Such numbers are called irrational; it was to simplify the writing of such numbers that the concept of a square root was introduced.

Let's look at another example to reinforce this. Let's look at the following problem: you need to cross a square field with a side of km diagonally, how many km do you have to go?

The most obvious thing here is to consider the triangle separately and use the Pythagorean theorem: . Thus, . So what is the required distance here? Obviously, the distance cannot be negative, we get that. The root of two is approximately equal, but, as we noted earlier, - is already a complete answer.

To solve examples with roots without causing problems, you need to see and recognize them. To do this, you need to know at least the squares of numbers from to, and also be able to recognize them. For example, you need to know what is equal to a square, and also, conversely, what is equal to a square.

Did you catch what a square root is? Then solve some examples.

Examples.

Well, how did it work out? Now let's look at these examples:

Answers:

Cube root

Well, we seem to have sorted out the concept of a square root, now let’s try to figure out what a cube root is and what is their difference.

The cube root of a number is the number whose cube is equal to. Have you noticed that everything is much simpler here? There are no restrictions on the possible values ​​of both the value under the cube root sign and the number being extracted. That is, the cube root can be extracted from any number: .

Do you understand what a cube root is and how to extract it? Then go ahead and solve the examples.

Examples.

Answers:

Root - oh degree

Well, we have understood the concepts of square and cube roots. Now let’s summarize the knowledge gained with the concept 1st root.

1st root of a number is a number whose th power is equal, i.e.

equivalent.

If - even, That:

• with negative, the expression does not make sense (even-th roots of negative numbers cannot be removed!);
• for non-negative() expression has one non-negative root.

If - is odd, then the expression has a unique root for any.

Don't be alarmed, the same principles apply here as with square and cube roots. That is, the principles that we applied when considering square roots are extended to all roots of even degree.

And the properties that were used for the cubic root apply to roots of odd degree.

Well, has it become clearer? Let's look at examples:

Here everything is more or less clear: first we look - yeah, the degree is even, the number under the root is positive, which means our task is to find a number whose fourth power will give us. Well, any guesses? Maybe, ? Exactly!

So, the degree is equal - odd, the number under the root is negative. Our task is to find a number that, when raised to a power, produces. It is quite difficult to immediately notice the root. However, you can immediately narrow your search, right? Firstly, the required number is definitely negative, and secondly, one can notice that it is odd, and therefore the desired number is odd. Try to find the root. Of course, you can safely dismiss it. Maybe, ?

Yes, this is what we were looking for! Note that to simplify the calculation we used the properties of degrees: .

Basic properties of roots

It's clear? If not, then after looking at the examples, everything should fall into place.

Multiplying roots

How to multiply roots? The simplest and most basic property helps answer this question:

Let's start with something simple:

Are the roots of the resulting numbers not exactly extracted? No problem - here are some examples:

What if there are not two, but more multipliers? The same! The formula for multiplying roots works with any number of factors:

What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!

Why do we need this? Yes, just to expand our capabilities when solving examples:

How do you like this property of roots? Does it make life much easier? For me, that's exactly right! You just have to remember that We can only enter positive numbers under the root sign of an even degree.

Let's see where else this can be useful. For example, the problem requires comparing two numbers:

What's more:

You can’t tell right away. Well, let's use the disassembled property of entering a number under the root sign? Then go ahead:

Well, knowing that the larger the number under the root sign, the larger the root itself! Those. if, then, . From this we firmly conclude that. And no one will convince us otherwise!

Before this, we entered a multiplier under the sign of the root, but how to remove it? You just need to factor it into factors and extract what you extract!

It was possible to take a different path and expand into other factors:

Not bad, right? Any of these approaches is correct, decide as you wish.

For example, here is an expression:

In this example, the degree is even, but what if it is odd? Again, apply the properties of powers and factor everything:

Everything seems clear with this, but how to extract the root of a number to a power? Here, for example, is this:

Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:

Well, is everything clear? Then here's an example:

These are the pitfalls, about them always worth remembering. This is actually reflected in the property examples:

It's clear? Reinforce with examples:

Yeah, we see that the root is to an even power, the negative number under the root is also to an even power. Well, does it work out the same? Here's what:

That's it! Now here are some examples:

Got it? Then go ahead and solve the examples.

Examples.

Answers.

If you have received answers, then you can move on with peace of mind. If not, then let's understand these examples:

Let's look at two other properties of roots:

These properties must be analyzed in examples. Well, let's do this?

Got it? Let's secure it.

Examples.

Answers.

ROOTS AND THEIR PROPERTIES. MIDDLE LEVEL

Arithmetic square root

The equation has two solutions: and. These are numbers whose square is equal to.

Consider the equation. Let's solve it graphically. Let's draw a graph of the function and a line at the level. The intersection points of these lines will be the solutions. We see that this equation also has two solutions - one positive, the other negative:

But in this case the solutions are not integers. Moreover, they are not rational. In order to write down these irrational decisions, we introduce a special square root symbol.

Arithmetic square root is a non-negative number whose square is equal to. When the expression is not defined, because There is no number whose square is equal to a negative number.

Square root: .

For example, . And it follows that or.

Let me draw your attention once again, this is very important: The square root is always a non-negative number: !

Cube root of a number is a number whose cube is equal to. The cube root is defined for everyone. It can be extracted from any number: . As we see, it can also take negative values.

The th root of a number is a number whose th power is equal, i.e.

If it is even, then:

• if, then the th root of a is not defined.
• if, then the non-negative root of the equation is called the arithmetic root of the th degree of and is denoted.

If - is odd, then the equation has a unique root for any.

Have you noticed that to the left above the sign of the root we write its degree? But not for the square root! If you see a root without a degree, it means it is square (degrees).

Examples.

Basic properties of roots

ROOTS AND THEIR PROPERTIES. BRIEFLY ABOUT THE MAIN THINGS

Square root (arithmetic square root) from a non-negative number is called this non-negative number whose square is

Properties of roots:

In this article we will introduce concept of a root of a number. We will proceed sequentially: we will start with the square root, from there we will move on to the description of the cubic root, after which we will generalize the concept of a root by defining the nth root. At the same time, we will introduce definitions, notations, 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, you need to 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

Square root of a is a number whose square is equal to a.

To lead 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 =5·5=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 given above, the number 5 is the square root of the number 25, the numbers −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 there exists a whose square is equal to a. Namely, for any negative number a there is no real number b whose square is equal to a. In fact, the equality a=b 2 is impossible for any negative a, since b 2 is a non-negative number for any b. Thus, there is no square root of a negative number on the set of real numbers. 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. This fact can be justified by the constructive method used to find the value of the square root.

Then the next 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's the answer: 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 of the number a is two, and the roots are . Let's justify this.

Let's start with the case a=0 . First, let's 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. Suppose there is some nonzero 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 arrived at 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. We said above that there is always a square root of any non-negative number, let the square root of a be the number b. Let's say that there is a number c, which is also the square root of a. Then, by the definition of a square root, the equalities b 2 =a and c 2 =a are true, 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 is valid properties of operations with real numbers possible only 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 is introduced 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.

The notation for the arithmetic square root of a is . The sign is called the arithmetic square root sign. It is also called the radical sign. Therefore, you can sometimes hear both “root” and “radical”, which means the same object.

The number under the arithmetic square root sign is called radical number, and the expression under the root sign is 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." The word “arithmetic” is used only when they want to emphasize that we are talking specifically about the positive square root of a number.

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

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 notation until we study complex numbers. For example, the expressions and are meaningless.

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

In conclusion of this point, 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 a number

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

Definition

Cube root of a is a number whose cube is equal to a.

Let's give 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 a 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 a number, unlike the square root, always exists, 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 square roots.

Moreover, there is only a single cube root of a given number a. Let us prove the last statement. 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 if a is positive, the cube root of a can be neither a negative number nor 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 another cube root of 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 possible only 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.

When a=0, the cube root of the number a is only the number 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, arguments similar to the case for positive a can be given. 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.

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

Let's give definition of arithmetic cube root.

Definition

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

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 index. The number under the root sign is radical 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 notations in which negative numbers are found under the sign of the arithmetic cube root. 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 point, let's say that the cube root of the number a is a solution of the form x 3 =a.

nth root, arithmetic root of degree n

Let us generalize the concept of a root of a number - we introduce definition of 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 first degree root of the number a is the number a itself, since when studying the degree with a natural exponent we took a 1 =a.

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

Let's start with the roots whose powers are the even numbers 4, 6, 8, ... As we already said, they are similar to the square root of the number a. That is, the root of any even degree of 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 even degree of the number a, and they are opposite numbers.

Let us substantiate the last statement. Let b be an even root (we denote it as 2·m, where m is some natural number) of the number a. Suppose that there is a number c - another root of degree 2·m from the number a. Then b 2·m −c 2·m =a−a=0 . But we know 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 on its left side there is 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 cubic root. That is, the root of any odd degree of the number a exists for any real number a, and for a given number a it is unique.

The uniqueness of a root of odd degree 2·m+1 of the number a is proved by analogy with the proof of the uniqueness of the cube root of 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 = is used (b−c)·(b 2·m +b 2·m−1 ·c+b 2·m−2 ·c 2 +… +c 2·m). The expression in the last bracket 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, with 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 in the highest nested parentheses is positive as the sum of the positive numbers. Now, moving sequentially to the expressions in brackets of the previous degrees of nesting, we are convinced that they are also positive as the sum 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 possible only when b−c=0, that is, when the number b is equal to the number c.

It's time to understand the notation of nth roots. For this purpose it is given definition of arithmetic root of the nth degree.

Definition

Arithmetic root of the nth degree of a non-negative number a is a non-negative number whose nth power is equal to a.

Let's solve a simple problem of finding the side of a square whose area is 9 cm 2. If we assume that the side of the square A cm, then we compose the equation according to the conditions of the problem:

A X A =9

A 2 =9

A 2 -9 =0

(A-3)(A+3)=0

A=3 or A=-3

The length of the side of a square cannot be a negative number, so the required side of the square is 3 cm.

When solving the equation, we found the numbers 3 and -3, the squares of which are 9. Each of these numbers is called the square root of the number 9. The non-negative of these roots, that is, the number 3, is called the arithmetic root of the number.

It is quite logical to accept the fact that the root can be found from numbers to the third power (cube root), fourth power, and so on. And in principle, the root is the inverse operation of exponentiation.

Rootn th degree from among α is such a number b, Where b n = α .

Here n- a natural number is usually called root index(or degree of root); as a rule, it is greater than or equal to 2, because the case n = 1 corny.

Designated on the letter as a symbol (root sign) on the right side is called radical. Number α - radical expression. For our example with a party, the solution could look like this: because (± 3) 2 = 9 .

We got the positive and negative values ​​of the root. This feature complicates calculations. To achieve unambiguity, the concept was introduced arithmetic root, the value of which is always with a plus sign, that is, only positive.

Root called arithmetic, if it is extracted from a positive number and is itself a positive number.

For example,

There is only one arithmetic root of a given degree from a given number.

The calculation operation is usually called “ root extraction n th degree" from among α . In essence, we perform the operation inverse to raising to a power, namely, finding the base of the power b according to a known indicator n and the result of raising to a power

α = bn.

The roots of the second and third degrees are used in practice more often than others and therefore they were given special names.

Square root: In this case, it is customary not to write the exponent 2, and the term “root” without indicating the exponent most often means the square root. Geometrically interpreted, is the length of the side of a square whose area is equal to α .

Cube root: Geometrically interpreted, the length of an edge of a cube whose volume is equal to α .

Properties of arithmetic roots.

1) When calculating arithmetic root of the product, it is necessary to extract it from each factor separately

For example,

2) For calculation root of a fraction, it is necessary to extract it from the numerator and denominator of this fraction

For example,

3) When calculating root of the degree, you need to divide the exponent by the root exponent

For example,

The first calculations related to extracting the square root were found in the works of mathematicians of ancient Babylon and China, India, Greece (there is no information in the sources about the achievements of ancient Egypt in this regard).

Mathematicians of ancient Babylon (2nd millennium BC) used a special numerical method to extract the square root. The initial approximation for the square root was found based on the natural number closest to the root (in the smaller direction) n. Presenting the radical expression in the form: α=n 2 +r, we get: x 0 =n+r/2n, then an iterative refinement process was applied:

The iterations in this method converge very quickly. For ,

For example, α=5; n=2; r=1; x 0 =9/4=2.25 and we get a sequence of approximations:

In the final value, all numbers are correct except the last one.

The Greeks formulated the problem of doubling the cube, which boiled down to constructing the cube root using a compass and ruler. The rules for calculating any degree of an integer have been studied by mathematicians in India and the Arab states. Then they were widely developed in medieval Europe.

Today, for the convenience of calculating square and cube roots, calculators are widely used.