or using the difference of squares formula like this:
- (x 2 -4)*(x 2 +4)=0.
The product of two factors is equal to zero if at least one of them is equal to zero.
The expression x 2 +4 cannot equal zero, therefore all that remains is (x 2 -4)=0.
We solve it and get two answers.
Answer: x=-2 and x=2.
We found that the equation x 4 =16 has only 2 real roots. These are roots of the fourth degree from the number 16. Moreover, the positive root is called the arithmetic root of the 4th degree from the number 16. And they are designated 4√16. That is, 4√16=2.
Definition
- An arithmetic root of a natural power n>=2 of a non-negative number a is some non-negative number, when raised to the power n, the number a is obtained.
It can be proven that for any non-negative a and natural n, the equation x n =a will have one single non-negative root. It is this root that is called the arithmetic root of the nth degree of the number a.
The arithmetic root of the nth degree of a number is denoted as follows: n√a.
The number a in this case is called a radical expression.
In the case when n=2, they do not write two, but simply write √a.
Arithmetic roots of the second and third degrees have their special names.
An arithmetic root of the second degree is called a square root, and an arithmetic root of the third degree is called a cube root.
Using just the definition of an arithmetic root, one can prove that n√a is equal to b. To do this we need to show that:
- 1. b is greater than or equal to zero.
- 2. b n =a.
For example, 3√(64) = 4, since 1. 4>0, 2. 4 3 =64.
Corollary to the definition of an arithmetic root.
- (n√a) n = a.
- n√(a n) = a.
For example, (5√2) 5 = 2.
Extracting the nth root
Extracting the nth root is the action that is used to find the nth root. Taking the nth root is the inverse of raising it to the nth power.
Let's look at an example.
Solve the equation x 3 = -27.
Let's rewrite this equation in the form (-x) 3 =27.
Let's put y=-x, then y 3 =27. This equation has one positive root y= 3√27 = 3.
This equation has no negative roots, since y 3
We find that the equation 3 =27 has only one root.
Returning to the original equation, we find that it also has only one root x=-y=-3.
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, 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.
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 =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 for finding the square root value.
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, we prove properties of square roots, 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 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 a general article. properties of roots.
Calculating the value of the cube root is called extracting the 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.
Root degree n from a real number a, Where n- natural number, such a real number is called x, n the th degree of which is equal to a.
Root degree n from the number 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 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.