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 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 not negative number for any b. Thus, on a set 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 justification for this fact can be considered constructive way, 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 ease of working with square roots negative root“separates” from the positive. For this purpose, it is introduced definition of arithmetic square root.

Definition

Arithmetic square root of a non-negative number a- This 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're talking about 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 . Arithmetic square root of zero equal to 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 cubic 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 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 c natural indicator we accepted 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 a root of even degree (we denote it as 2 m, where m is some natural number) from 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, any root odd degree from 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 parentheses itself high degree nesting, is positive as the sum of 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 arithmetic root 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.

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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 given 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 special concept 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 original equation will give the correct result. In our quadratic equation suitable for both.

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 memorize it, you need to remember (or be able to quickly figure it out) 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 possible values both the values ​​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 what larger number under the sign of the root, 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 in this case 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:

Arithmetic root of the second degree

Definition 1

The second root (or square root) of $a$ call a number that, when squared, becomes equal to $a$.

Example 1

$7^2=7 \cdot 7=49$, which means the number $7$ is the 2nd root of the number $49$;

$0.9^2=0.9 \cdot 0.9=0.81$, which means the number $0.9$ is the 2nd root of the number $0.81$;

$1^2=1 \cdot 1=1$, which means the number $1$ is the 2nd root of the number $1$.

Note 2

Simply put, for any number $a

$a=b^2$ for negative $a$ is incorrect, because $a=b^2$ cannot be negative for any value of $b$.

It can be concluded that for real numbers there cannot be a 2nd root of a negative number.

Note 3

Because $0^2=0 \cdot 0=0$, then from the definition it follows that zero is the 2nd root of zero.

Definition 2

Arithmetic root of the 2nd degree of the number $a$($a \ge 0$) is a non-negative number that, when squared, equals $a$.

Roots of the 2nd degree are also called square roots.

The arithmetic root of the 2nd degree of the number $a$ is denoted as $\sqrt(a)$ or you can see the notation $\sqrt(a)$. But most often for the square root the number $2$ is root exponent– not specified. The sign “$\sqrt( )$” is the sign of the arithmetic root of the 2nd degree, which is also called “ radical sign" The concepts “root” and “radical” are names of the same object.

If there is a number under the arithmetic root sign, then it is called radical number, and if the expression, then – radical expression.

The entry $\sqrt(8)$ is read as “arithmetic root of the 2nd degree of eight,” and the word “arithmetic” is often not used.

Definition 3

According to definition arithmetic root of the 2nd degree can be written:

For any $a \ge 0$:

$(\sqrt(a))^2=a$,

$\sqrt(a)\ge 0$.

We showed the difference between a second root and an arithmetic second root. Further we will consider only roots of non-negative numbers and expressions, i.e. only arithmetic.

Arithmetic root of the third degree

Definition 4

Arithmetic root of the 3rd degree (or cube root) of the number $a$($a \ge 0$) is a non-negative number that, when cubed, becomes equal to $a$.

Often the word arithmetic is omitted and they say “the 3rd root of the number $a$”.

The arithmetic root of the 3rd degree of $a$ is denoted as $\sqrt(a)$, the sign “$\sqrt( )$” is the sign of the arithmetic root of the 3rd degree, and the number $3$ in this notation is called root index. The number or expression that appears under the root sign is called radical.

Example 2

$\sqrt(3,5)$ – arithmetic root of the 3rd degree of $3.5$ or cube root of $3.5$;

$\sqrt(x+5)$ – arithmetic root of the 3rd degree of $x+5$ or cube root of $x+5$.

Arithmetic nth root

Definition 5

Arithmetic nth root degrees from the number $a \ge 0$ a non-negative number is called which, when raised to the $n$th power, becomes equal to $a$.

Notation for the arithmetic root of degree $n$ of $a \ge 0$:

where $a$ is a radical number or expression,

An arithmetic root of the nth degree of a non-negative number is a non-negative number nth degree which 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 not an integer even number (), 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 given 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 - Not even 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.