Determination of the root degree of a real. Root of degree n: basic definitions. Root of the nth degree. Definition

Lesson and presentation on the topic: "The nth root of a real number"

Additional materials
Dear users, do not forget to leave your comments, reviews, wishes! All materials have been checked by an anti-virus program.

Teaching aids and simulators in the Integral online store for grade 11
Algebraic problems with parameters, grades 9–11
"Interactive tasks on building in space for grades 10 and 11"

Root of the nth degree. Repetition of what has been covered.

Guys, the topic of today's lesson is called "Nth root of a real number".
We studied the square root of a real number in 8th grade. The square root is related to a function of the form $y=x^2$. Guys, do you remember how we calculated square roots, and what properties did it have? Repeat this topic yourself.
Let's look at a function of the form $y=x^4$ and plot it.

Now let's solve the equation graphically: $x^4=16$.
Let's draw a straight line $y=16$ on our graph of the function and see at what points our two graphs intersect.
The graph of the function clearly shows that we have two solutions. The functions intersect at two points with coordinates (-2;16) and (2;16). The abscissas of our points are the solutions to our equation: $x_1=-2$ and $x_2=2$. It is also easy to find the roots of the equation $x^4=1$; obviously, $x_1=-1$ and $x_2=1$.
What to do if there is an equation $x^4=7$.
Let's plot our functions:
Our graph clearly shows that the equation also has two roots. They are symmetrical about the ordinate axis, that is, they are opposite. It is not possible to find an exact solution from the graph of functions. We can only say that our solutions are modulo less than 2 but greater than 1. We can also say that our roots are irrational numbers.
Faced with such a problem, mathematicians needed to describe it. They introduced a new notation: $\sqrt()$, which they called the fourth root. Then the roots of our equation $x^4=7$ will be written in this form: $x_1=-\sqrt(7)$ and $x_2=\sqrt(7)$. Read as the fourth root of seven.
We talked about an equation of the form $x^4=a$, where $a>0$ $(a=1,7,16)$. We can consider equations of the form: $x^n=a$, where $a>0$, n - any natural number.
We should pay attention to the degree at x, whether the degree is even or odd - the number of solutions changes. let's consider specific example. Let's solve the equation $x^5=8$. Let's plot the function:
The graph of the functions clearly shows that in our case we have only one solution. The solution is usually denoted as $\sqrt(8)$. Solving an equation of the form $x^5=a$ and running along the entire ordinate axis, it is not difficult to understand that this equation will always have one solution. In this case, the value of a may be less than zero.

Root of the nth degree. Definition

Definition. The nth root ($n=2,3,4...$) of a non-negative number a is called non-negative number, when raised to the power n, the number a is obtained.

This number is denoted as $\sqrt[n](a)$. The number a is called the radical number, n is the root exponent.

Roots of the second and third degrees are usually called square and cubic roots, respectively. We studied them in eighth and ninth grade.
If $а≥0$, $n=2,3,4,5…$, then:
1) $\sqrt[n](a)≥0,$
2) $(\sqrt[n](a))^n=a.$
The operation of finding the root of a non-negative number is called "root extraction".
Exponentiation and root extraction are the same dependency:

Guys, please note that the table contains only positive numbers. In the definition, we stipulated that the root is taken only from a non-negative number a. Next we will clarify when it is possible to extract the root of a negative number a.

Root of the nth degree. Examples of solutions

Calculate:
a) $\sqrt(64)$.
Solution: $\sqrt(64)=8$, since $8>0$ and $8^2=64$.

B) $\sqrt(0.064)$.
Solution: $\sqrt(0.064)=0.4$, since $0.4>0$ and $0.4^3=0.064$.

B) $\sqrt(0)$.
Solution: $\sqrt(0)=0$.

D) $\sqrt(34)$.
Solution: In this example, we cannot find out the exact value, our number is irrational. But we can say that it is greater than 2 and less than 3, since 2 to the 5th power is equal to 32, and 3 to the 5th power is equal to 243. 34 lies between these numbers. We can find an approximate value using a calculator that can calculate the roots of $\sqrt(34)≈2.02$ with an accuracy of thousandths.
In our definition, we agreed to calculate nth roots only from positive numbers. At the beginning of the lesson, we saw an example that it is possible to extract nth roots from negative numbers. We have looked at the odd exponent of the function and now let's make some clarifications.

Definition. A root of an odd power n (n=3,5,7,9...) of a negative number a is a negative number such that when raised to the power n, the result is a.

It is customary to use the same designations.
If $a 1) $\sqrt[n](a) 2) $(\sqrt[n](a))^n=a$.
An even root makes sense only for a positive radical number; an odd root makes sense for any radical number.

Examples.
a) Solve the equations: $\sqrt(3x+3)=-3$.
Solution: If $\sqrt(y)=-3$, then $y=-27$. That is, both sides of our equation must be cubed.
$3x+3=-27$.
$3x=-30$.
$x=-10$.

B) Solve the equations: $\sqrt(2x-1)=1$.
Let's raise both sides to the fourth power:
$2x-1=1$.
$2х=2$.
$x=1$.

C) Solve the equations: $\sqrt(4x-1)=-5$.
Solution: According to our definition, a root of an even degree can only be taken from a positive number, but we are given a negative number, then there are no roots.

D) Solve the equations: $\sqrt(x^2-7x+44)=2$.
Solution: Raise both sides of the equation to the fifth power:
$x^2-7x+44=32$.
$x^2-7x+12=0$.
$x_1=4$ and $x_2=3$.

Problems to solve independently

1. Calculate:
a) $\sqrt(81)$.
b) $\sqrt(0.0016)$.
c) $\sqrt(1)$.
d) $\sqrt(70)$.
2. Solve the equations:
a) $\sqrt(2x+6)=2$.
b) $\sqrt(3x-5)=-1$.
c) $\sqrt(4x-8)=-4$.
d) $\sqrt(x^2-8x+49)=2$.

X 4 =1 and solve it graphically. To do this, in one coordinate system we will construct a graph of the function y = x n straight line y = 1 (Fig. 164 a). They intersect at two points:

They are the roots of the equation x 4 = 1.
Reasoning in exactly the same way, we find the roots of the equation x 4 = 16:

Now let's try to solve the equation x 4 =5; a geometric illustration is shown in Fig. 164 b. It is clear that the equation has two roots x 1 and x 2, and these numbers, as in the two previous cases, are mutually opposite. But for the first two equations roots were found without difficulty (they could be found without using graphs), but there are problems with the equation x 4 = 5: according to the drawing, we cannot indicate the values of the roots, but can only establish that one root is located to the left of the point -1, and the second - to the right of point 1.
It can be proven (in much the same way as was done in our textbook “Algebra-8” for the number l/b) that x 1 and x 2 are irrational numbers (i.e., infinite non-periodic decimal fractions).

Having encountered a similar situation for the first time, mathematicians realized that they needed to come up with a way to describe it in terms of mathematical language. They introduced a new symbol, which they called the fourth root, and using this symbol, the roots of the equation x 4 = 5 were written as follows: (read: “fourth root of five”).

Note 1. Compare these arguments with similar arguments carried out in § 17, 32 and 38. New terms and new notations in mathematics appear when they are needed to describe a new mathematical models. This is a reflection of the peculiarities of mathematical language: its main function is not communicative - for communication, but organizing - for organization successful work with mathematical models in different areas knowledge.

We talked about the equation x 4 = a, where a > 0. With equal success we could talk about the equation x 4 = a, where a > 0, and n is any natural number. For example, solving graphically the equation x 5 = 1, we find x = 1 (Fig. 165); solving the equation x 5 " = 7, we establish that the equation has one root xr, which is located on the x axis slightly to the right of point 1 (see Fig. 165). For the number xx, we introduce the notation Hh.

In general, solving the equation x n = a, where a > 0, n e N, n > 1, in the case of even n we obtain two roots: (Fig. 164, c); in the case of odd n - one root (read: “root nth degree from number a"). Solving the equation x n =0, we obtain the only root x = 0.

Note 2. In mathematical language, as in ordinary language, it happens that the same term is applied to different concepts; Thus, in the previous sentence the word “root” is used in two senses: as the root of an equation (you have long been accustomed to this interpretation) and as the root lth degree from the number (new interpretation). It is usually clear from the context what interpretation of the term is intended.

Now we are ready to give a precise definition.

Definition 1. Root lth powers of a non-negative number a (n = 2, 3,4, 5,...) is a non-negative number that, when raised to a power n, results in the number a.

This number is denoted, the number a is called the radical number, and the number n is the exponent of the root.
If n=2, then they usually don’t say “second root,” but say “square root.” In this case, they don’t write This is the one special case, which you specifically studied in your 8th grade algebra course.

If n = 3, then instead of “third degree root” they often say “cube root”. Your first acquaintance with the cube root also took place in the 8th grade algebra course. We used the cube root in §36 to solve Example 6.

In general, it’s the same mathematical model(the same relationship between non-negative numbers a and b), but only the second one is described more in simple language(uses simpler characters) than the first.

The operation of finding the root of a non-negative number is usually called root extraction. This operation is the reverse of raising to the appropriate power. Compare:

Sometimes the expression is called a radical (from the Latin word gadix - “root”). In Russian, the term radical is used quite often, for example, “radical changes” - this means “radical changes”. By the way, the very designation of the root is reminiscent of the word gadix: the symbol is a stylized letter r.

Example 1. Calculate:

d) Unlike previous examples, we cannot indicate the exact value of the number. It is only clear that it is greater than 2, but less than 3, since 2 4 = 16 (this is less than 17), and 3 4 = 81 (this more than 17). We note that 24 is much closer to 17 than 34, so there is reason to use the approximate equality sign:

However, a more accurate approximate value of a number can be found using a calculator that contains the operation of extracting the root; it is approximately equal to
The operation of extracting the root is also determined for a negative radical number, but only in the case of an odd root exponent. In other words, the equality (-2)5 = -32 can be rewritten in equivalent form as . The following definition is used.

Definition 2. An odd root n of a negative number a (n = 3.5,...) is a negative number that, when raised to the power n, results in the number a.

This number, as in Definition 1, is denoted by , the number a is the radical number, and the number n is the exponent of the root.
So,

Thus, an even root has meaning (i.e., is defined) only for a non-negative radical expression; an odd root makes sense for any radical expression.
Example 2. Solve equations:

Solution: and if Actually both parts given equation we must cube. We get:

b) Reasoning as in example a), we raise both sides of the equation to the fourth power. We get:

c) There is no need to raise it to the fourth power; this equation has no solutions. Why? Because, according to definition 1, an even root is a non-negative number.
d) Raising both sides of the equation to the sixth power, we get:

A.G. Mordkovich Algebra 10th grade

Lesson content lesson notes supporting frame lesson presentation acceleration methods interactive technologies Practice tasks and exercises self-test workshops, trainings, cases, quests homework discussion questions rhetorical questions from students Illustrations audio, video clips and multimedia photographs, pictures, graphics, tables, diagrams, humor, anecdotes, jokes, comics, parables, sayings, crosswords, quotes Add-ons abstracts articles tricks for the curious cribs textbooks basic and additional dictionary of terms other Improving textbooks and lessonscorrecting errors in the textbook updating a fragment in a textbook, elements of innovation in the lesson, replacing outdated knowledge with new ones Only for teachers perfect lessons calendar plan for a year guidelines discussion programs Integrated Lessons

Lesson script for 11th grade on the topic:

“The nth root of a real number. »

The purpose of the lesson: Formation in students of a holistic understanding of the root n-th degree and arithmetic root of the nth degree, formation of computational skills, conscious and rational use properties of the root when solving various problems containing a radical. Check the level of students' understanding of the topic's questions.

Subject:create meaningful and organizational conditions for mastering material on the topic “ Numeric and alphabetic expressions » at the level of perception, comprehension and primary memorization; develop the ability to use this information when calculating nth root powers from a real number;

Meta-subject: promote the development of computing skills; ability to analyze, compare, generalize, draw conclusions;

Personal: cultivate the ability to express one’s point of view, listen to the answers of others, take part in dialogue, and develop the ability for positive cooperation.

Planned result.

Subject: be able to apply in a real situation the properties of the nth root of a real number when calculating roots and solving equations.

Personal: to develop attentiveness and accuracy in calculations, a demanding attitude towards oneself and one’s work, and to cultivate a sense of mutual assistance.

Lesson type: lesson on studying and initially consolidating new knowledge

Motivation for educational activities:

Eastern wisdom says: “You can lead a horse to water, but you cannot force him to drink.” And it is impossible to force a person to study well if he himself does not try to learn more, does not have the desire to work on his mental development. After all, knowledge is only knowledge when it is acquired through the efforts of one’s thoughts, and not through memory alone.

Our lesson will be held under the motto: “We will conquer any peak if we strive for it.” During the lesson, you and I need to have time to overcome several peaks, and each of you must put all your efforts into conquering these peaks.

“Today we have a lesson in which we must get acquainted with a new concept: “Nth root” and learn how to apply this concept to the transformation of various expressions.

Your goal is to activate your existing knowledge through various forms of work, contribute to the study of the material and get good grades.”
We studied the square root of a real number in 8th grade. The square root is related to a function of the form y=x 2. Guys, do you remember how we calculated square roots, and what properties did it have?
a) individual survey:

what kind of expression is this

what is called square root

what is called arithmetic square root

list the properties of square root

b) work in pairs: calculate.

2. Updating knowledge and creating a problem situation: Solve the equation x 4 =1. How can we solve it? (Analytical and graphical). Let's solve it graphically. To do this, in one coordinate system we will construct a graph of the function y = x 4 straight line y = 1 (Fig. 164 a). They intersect at two points: A (-1;1) and B(1;1). Abscissas of points A and B, i.e. x 1 = -1,

x 2 = 1 are the roots of the equation x 4 = 1.
Reasoning in exactly the same way, we find the roots of the equation x 4 =16: Now let’s try to solve the equation x 4 =5; a geometric illustration is shown in Fig. 164 b. It is clear that the equation has two roots x 1 and x 2, and these numbers, as in the two previous cases, are mutually opposite. But for the first two equations the roots were found without difficulty (they could be found without using graphs), but with the equation x 4 = 5 there are problems: from the drawing we cannot indicate the values of the roots, but we can only establish that one root is located to the left point -1, and the second one is to the right of point 1.

x 2 = - (read: “fourth root of five”).

We talked about the equation x 4 = a, where a 0. We could equally well talk about the equation x 4 = a, where a 0, and n is any natural number. For example, solving graphically the equation x 5 = 1, we find x = 1 (Fig. 165); solving the equation x 5 "= 7, we establish that the equation has one root x 1, which is located on the x axis slightly to the right of point 1 (see Fig. 165). For the number x 1, we introduce the notation .

Definition 1. The nth root of a non-negative number a (n = 2, 3,4, 5,...) is a non-negative number that, when raised to the power n, results in the number a.

This number is denoted, the number a is called the radical number, and the number n is the exponent of the root.
If n=2, then they usually don’t say “second root,” but say “square root.” In this case, they don’t write this. This is the special case that you specifically studied in the 8th grade algebra course.

If n = 3, then instead of “third degree root” they often say “cube root”. Your first acquaintance with the cube root also took place in the 8th grade algebra course. We used cube roots in 9th grade algebra.

So, if a ≥0, n= 2,3,4,5,…, then 1) ≥ 0; 2) () n = a.

In general, =b and b n =a are the same relationship between non-negative numbers a and b, but only the second is described in a simpler language (uses simpler symbols) than the first.

The operation of finding the root of a non-negative number is usually called root extraction. This operation is the reverse of raising to the appropriate power. Compare:

Please note again: only positive numbers appear in the table, since this is stipulated in Definition 1. And although, for example, (-6) 6 = 36 is a correct equality, go from it to notation using the square root, i.e. write that it is impossible. By definition, a positive number means = 6 (not -6). In the same way, although 2 4 =16, t (-2) 4 =16, moving to the signs of the roots, we must write = 2 (and at the same time ≠-2).

The operation of extracting the root is also determined for a negative radical number, but only in the case of an odd root exponent. In other words, the equality (-2) 5 = -32 can be rewritten in equivalent form as =-2. The following definition is used.

Definition 2. An odd root n of a negative number a (n = 3.5,...) is a negative number that, when raised to the power n, results in the number a.

This number, as in Definition 1, is denoted by , the number a is the radical number, and the number n is the exponent of the root.
So, if a , n=,5,7,…, then: 1) 0; 2) () n = a.

Thus, an even root has meaning (i.e., is defined) only for a non-negative radical expression; an odd root makes sense for any radical expression.

5. Primary consolidation of knowledge:

1. Calculate: No. 33.5; 33.6; 33.74 33.8 orally a) ; b) ; V) ; G) .

Place the corresponding letter next to the example.

A little information about the great scientist. Rene Descartes (1596-1650) French nobleman, mathematician, philosopher, physiologist, thinker. Rene Descartes laid the foundations analytical geometry, entered the letter designations x 2, y 3. Everyone knows Cartesian coordinates, defining the function of the variable.

3 . Solve the equations: a) = -2; b) = 1; c) = -4

Solution: a) If = -2, then y = -8. In fact, we must cube both sides of the given equation. We get: 3x+4= - 8; 3x= -12; x = -4. b) Reasoning as in example a), we raise both sides of the equation to the fourth power. We get: x=1.

c) There is no need to raise it to the fourth power; this equation has no solutions. Why? Because, according to definition 1, an even root is a non-negative number.
Several tasks are offered to your attention. When you complete these tasks, you will learn the name and surname of the great mathematician. This scientist was the first to introduce the root sign in 1637.

6. Let's have a little rest.

The class raises its hands - this is “one”.

The head turned - it was “two”.

Hands down, look forward - this is “three”.

Hands turned wider to the sides to “four”

Pressing them with force into your hands is a “high five.”

All the guys need to sit down - it’s “six”.

7. Independent work:

option: option 2:

b) 3-. b)12 -6.

2. Solve the equation: a) x 4 = -16; b) 0.02x 6 -1.28=0; a) x 8 = -3; b)0.3x 9 – 2.4=0;

c) = -2; c)= 2

8. Repetition: Find the root of the equation = - x. If the equation has more than one root, write the answer with the smaller root.

9. Reflection: What did you learn in the lesson? What was interesting? What was difficult?

To successfully use the root extraction operation in practice, you need to become familiar with the properties of this operation.
All properties are formulated and proven only for non-negative values of the variables contained under the signs of the roots.

Theorem 1. The nth root (n=2, 3, 4,...) of the product of two non-negative chips is equal to the product nth roots powers of these numbers:

Comment:

1. Theorem 1 remains valid for the case when the radical expression is the product of more than two non-negative numbers.

Theorem 2.If, and n is a natural number greater than 1, then the equality is true

Brief(albeit inaccurate) formulation, which is more convenient to use in practice: the root of a fraction is equal to the fraction of the roots.

Theorem 1 allows us to multiply t only roots of the same degree , i.e. only roots with the same index.

Theorem 3.If ,k is a natural number and n is a natural number greater than 1, then the equality is true

In other words, to build a root in natural degree, it is enough to raise the radical expression to this power.
This is a consequence of Theorem 1. In fact, for example, for k = 3 we obtain: We can reason in exactly the same way in the case of any other natural value of the exponent k.

Theorem 4.If ,k, n are natural numbers greater than 1, then the equality is true

In other words, to extract a root from a root, it is enough to multiply the indicators of the roots.
For example,

Be careful! We learned that four operations can be performed on roots: multiplication, division, exponentiation, and root extraction (from the root). But what about adding and subtracting roots? No way.
For example, instead of writing Really, But it’s obvious that

Theorem 5.If the indicators of the root and radical expression are multiplied or divided by the same natural number, then the value of the root will not change, i.e.

Examples of problem solving

Example 1. Calculate

Solution. Using the first property of roots (Theorem 1), we obtain:

Example 2. Calculate
Solution. Let's reverse mixed number into an improper fraction.
We have Using the second property of roots ( Theorem 2 ), we get:

Example 3. Calculate:

Solution. Any formula in algebra, as you well know, is used not only “from left to right”, but also “from right to left”. Thus, the first property of roots means that they can be represented in the form and, conversely, can be replaced by the expression. The same applies to the second property of roots. Taking this into account, let's perform the calculations.

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 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'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 . 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 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 under the sign of the arithmetic cube root are negative numbers. 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 the roots of even powers are similar square root, and roots of odd powers - cubic. Let's deal with them one by one.

Let's start with the roots, the powers of which are even numbers 4, 6, 8, ... As we 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 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 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.