Multiplication and division of the nth root. Power function and roots - definition, properties and formulas. Equation x n =a

Lesson objectives:

Educational: create conditions for the formation in students of a holistic idea of ​​the nth root, skills of conscious and rational use of the properties of the root when solving various problems.

Developmental: create conditions for the development of algorithmic, creative thinking, develop self-control skills.

Educational: promote the development of interest in the subject, activity, cultivate accuracy in work, the ability to express one’s own opinion, and give recommendations.

Lesson progress

1. Organizational moment.

Good afternoon Good hour!

I'm so glad to see you.

The bell has already rung

The lesson begins.

We smiled. We caught up.

We looked at each other

And they sat down quietly together.

2. Lesson motivation.

The outstanding French philosopher and scientist Blaise Pascal argued: “The greatness of a person is in his ability to think.” Today we will try to feel like great people by discovering knowledge for ourselves. The motto for today's lesson will be the words of the ancient Greek mathematician Thales:

What is there more than anything in the world? - Space.

What's the fastest? - Mind.

What is the wisest thing? - Time.

What's the best part? - Achieve what you want.

I would like each of you to achieve the desired result in today’s lesson.

3. Updating knowledge.

1. Name the reciprocal algebraic operations on numbers. (Addition and subtraction, multiplication and division)

2. Is it always possible to perform an algebraic operation such as division? (No, you cannot divide by zero)

3. What other operation can you perform with numbers? (Exponentiation)

4. What operation will be her reverse? (Root extraction)

5. What degree of root can you extract? (Second root)

6. What properties of the square root do you know? (Extracting the square root of a product, from a quotient, from a root, raising to a power)

7. Find the meanings of the expressions:

From history. As early as 4,000 years ago, Babylonian scientists compiled, along with multiplication tables and reciprocal tables (with the help of which division of numbers was reduced to multiplication), tables of squares of numbers and square roots of numbers. At the same time, they were able to find the approximate value of the square root of any integer.

4. Studying new material.

Obviously, in accordance with the basic properties of powers with natural exponents, from any positive number there are two opposite values ​​of the root of an even power, for example, the numbers 4 and -4 are square roots of 16, since (-4) 2 = 42 = 16, and the numbers 3 and -3 are the fourth roots of 81, since (-3)4 = 34 = 81.

Moreover, there is no even root of a negative number, since the even power of any real number is non-negative. As for the root of an odd degree, for any real number there is only one root of an odd degree from this number. For example, 3 is the third root of 27, since 33 = 27, and -2 is the fifth root of -32, since (-2)5 = 32.

Due to the existence of two roots of even degree from a positive number, we introduce the concept of an arithmetic root in order to eliminate this ambiguity of the root.

A non-negative value of the nth root of a non-negative number is called an arithmetic root.

Designation: - root of the nth degree.

The number n is called the power of the arithmetic root. If n = 2, then the degree of the root is not indicated and is written. The root of the second degree is usually called the square root, and the root of the third degree is called the cubic root.

B, b2 = a, a ≥ 0, b ≥ 0

B, bп = a, p - even a ≥ 0, b ≥ 0

n - odd a, b - any

Properties

1. , a ≥ 0, b ≥ 0

2. , a ≥ 0, b >0

3. , a ≥ 0

4. , m, n, k - natural numbers

5. Consolidation of new material.

Oral work

a) Which expressions make sense?

b) For what values ​​of the variable a does the expression make sense?

Solve No. 3, 4, 7, 9, 11.

6. Physical education minute.

Moderation is needed in all matters,

Let it be the main rule.

Do gymnastics, since you’ve been thinking for a long time,

Gymnastics does not exhaust the body,

But it cleanses the body completely!

Close your eyes, relax your body,

Imagine - you are birds, you suddenly fly!

Now you are swimming in the ocean like a dolphin,

Now you are picking ripe apples in the garden.

Left, right, looked around,

Open your eyes and get back to business!

7. Independent work.

Work in pairs with. 178 No. 1, No. 2.

8. D/z. Learn item 10 (p. 160-161), solve No. 5, 6, 8, 12, 16 (1, 2).

9. Lesson summary. Reflection of activity.

Did the lesson achieve its goal?

What have you learned?

Lesson script for 11th grade on the topic:

“The nth root of a real number. »

Objective 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, skills of conscious and rational use of the 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 the nth root of 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 and 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 in all your efforts to conquer 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 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 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).

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.

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) .

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:
2. Find the meanings of the following expressions.

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 of analytical geometry and introduced the letter designations x 2, y 3. Everyone knows the Cartesian coordinates that define a function of a 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?

Lesson and presentation on the topic: "Properties of the nth root. Theorems"

Additional materials
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Teaching aids and simulators in the Integral online store for grade 11
Interactive manual for grades 9–11 "Trigonometry"
Interactive manual for grades 10–11 "Logarithms"

Properties of the nth root. Theorems

Guys, we continue to study the nth roots of a real number. Like almost all mathematical objects, roots of the nth degree have certain properties, today we will study them.
All the properties that we will consider are formulated and proven only for non-negative values ​​of the variables contained under the root sign.
In the case of an odd root exponent, they are also performed for negative variables.

Theorem 1. The nth root of the product of two non-negative numbers is equal to the product of the nth roots of these numbers: $\sqrt[n](a*b)=\sqrt[n](a)*\sqrt[n]( b)$ .

Let's prove the theorem.
Proof. Guys, to prove the theorem, let's introduce new variables, denote them:
$\sqrt[n](a*b)=x$.
$\sqrt[n](a)=y$.
$\sqrt[n](b)=z$.
We need to prove that $x=y*z$.
Note that the following identities also hold:
$a*b=x^n$.
$a=y^n$.
$b=z^n$.
Then the following identity holds: $x^n=y^n*z^n=(y*z)^n$.
The powers of two non-negative numbers and their exponents are equal, then the bases of the powers themselves are equal. This means $x=y*z$, which is what needed to be proven.

Theorem 2. If $a≥0$, $b>0$ and n is a natural number greater than 1, then the following equality holds: $\sqrt[n](\frac(a)(b))=\frac(\sqrt[ n](a))(\sqrt[n](b))$.

That is, the nth root of the quotient is equal to the quotient of the nth roots.

Proof.
To prove this, we will use a simplified diagram in the form of a table:

Examples of calculating the nth root

Example.
Calculate: $\sqrt(16*81*256)$.
Solution. Let's use Theorem 1: $\sqrt(16*81*256)=\sqrt(16)*\sqrt(81)*\sqrt(256)=2*3*4=24$.

Example.
Calculate: $\sqrt(7\frac(19)(32))$.
Solution. Let's imagine the radical expression as an improper fraction: $7\frac(19)(32)=\frac(7*32+19)(32)=\frac(243)(32)$.
Let's use Theorem 2: $\sqrt(\frac(243)(32))=\frac(\sqrt(243))(\sqrt(32))=\frac(3)(2)=1\frac(1) (2)$.

Example.
Calculate:
a) $\sqrt(24)*\sqrt(54)$.
b) $\frac(\sqrt(256))(\sqrt(4))$.
Solution:
a) $\sqrt(24)*\sqrt(54)=\sqrt(24*54)=\sqrt(8*3*2*27)=\sqrt(16*81)=\sqrt(16)*\ sqrt(81)=2*3=6$.
b) $\frac(\sqrt(256))(\sqrt(4))=\sqrt(\frac(256)(4))=\sqrt(64)=24$.

Theorem 3. If $a≥0$, k and n are natural numbers greater than 1, then the equality holds: $(\sqrt[n](a))^k=\sqrt[n](a^k)$.

To raise a root to a natural power, it is enough to raise the radical expression to this power.

Proof.
Let's look at the special case for $k=3$. Let's use Theorem 1.
$(\sqrt[n](a))^k=\sqrt[n](a)*\sqrt[n](a)*\sqrt[n](a)=\sqrt[n](a*a *a)=\sqrt[n](a^3)$.
The same can be proven for any other case. Guys, prove it yourself for the case when $k=4$ and $k=6$.

Theorem 4. If $a≥0$ b n,k are natural numbers greater than 1, then the equality holds: $\sqrt[n](\sqrt[k](a))=\sqrt(a)$.

To extract a root from a root, it is enough to multiply the indicators of the roots.

Proof.
Let us prove it briefly again using a table. To prove this, we will use a simplified diagram in the form of a table:

Example.
$\sqrt(\sqrt(a))=\sqrt(a)$.
$\sqrt(\sqrt(a))=\sqrt(a)$.
$\sqrt(\sqrt(a))=\sqrt(a)$.

Theorem 5. If the exponents of the root and radical expression are multiplied by the same natural number, then the value of the root will not change: $\sqrt(a^(kp))=\sqrt[n](a)$.

Proof.
The principle of proving our theorem is the same as in other examples. Let's introduce new variables:
$\sqrt(a^(k*p))=x=>a^(k*p)=x^(n*p)$ (by definition).
$\sqrt[n](a^k)=y=>y^n=a^k$ (by definition).
Let us raise the last equality to the power p
$(y^n)^p=y^(n*p)=(a^k)^p=a^(k*p)$.
Received:
$y^(n*p)=a^(k*p)=x^(n*p)=>x=y$.
That is, $\sqrt(a^(k*p))=\sqrt[n](a^k)$, which is what needed to be proven.

Examples:
$\sqrt(a^5)=\sqrt(a)$ (divided the indicators by 5).
$\sqrt(a^(22))=\sqrt(a^(11))$ (divided the indicators by 2).
$\sqrt(a^4)=\sqrt(a^(12))$ (indicators multiplied by 3).

Example.
Perform actions: $\sqrt(a)*\sqrt(a)$.
Solution.
The exponents of the roots are different numbers, so we cannot use Theorem 1, but by applying Theorem 5, we can get equal exponents.
$\sqrt(a)=\sqrt(a^3)$ (indicators multiplied by 3).
$\sqrt(a)=\sqrt(a^4)$ (indicators multiplied by 4).
$\sqrt(a)*\sqrt(a)=\sqrt(a^3)*\sqrt(a^4)=\sqrt(a^3*a^4)=\sqrt(a^7)$.

Problems to solve independently

1. Calculate: $\sqrt(32*243*1024)$.
2. Calculate: $\sqrt(7\frac(58)(81))$.
3. Calculate:
a) $\sqrt(81)*\sqrt(72)$.
b) $\frac(\sqrt(1215))(\sqrt(5))$.
4. Simplify:
a) $\sqrt(\sqrt(a))$.
b) $\sqrt(\sqrt(a))$.
c) $\sqrt(\sqrt(a))$.
5. Perform actions: $\sqrt(a^2)*\sqrt(a^4)$.

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Rootn-th degree and its properties

What is a rootnth degree? How to extract the root?

In eighth grade, you have already become acquainted with square root. We solved typical examples with roots, using certain properties of roots. Also decided quadratic equations, where without extracting the square root - no way. But the square root is just a special case of a broader concept - root n th degree . In addition to the square root, there are, for example, cube roots, fourth, fifth and higher powers. And to successfully work with such roots, it would be a good idea to first be on familiar terms with square roots.) Therefore, anyone who has problems with them, I strongly recommend repeating this.

Extracting the root is one of the operations inverse to raising to a power.) Why “one of”? Because when we extract the root, we are looking for base according to known degree and indicator. And there is another inverse operation - finding indicator according to known degree and basis. This operation is called finding logarithm It is more complex than root extraction and is studied in high school.)

So, let's get acquainted!

First, the designation. The square root, as we already know, is denoted like this: . This icon is called very beautifully and scientifically - radical. What are the roots of other degrees? It’s very simple: above the “tail” of the radical, additionally write the exponent of the degree whose root is being sought. If you are looking for a cube root, then write a triple: . If the root is of the fourth degree, then, accordingly, . And so on.) In general, the nth root is denoted like this:

Where .

Numbera , as in square roots, is called radical expression , and here is the numbern This is new for us. And it's called root index .

How to extract roots of any degrees? Just like square ones - figure out what number to the nth power gives us the numbera .)

How, for example, do you take the cube root of 8? That is ? What number cubed will give us 8? A deuce, naturally.) So they write:

Or . What number to the fourth power gives 81? Three.) So,

What about the tenth root of 1? Well, it’s a no brainer that one to any power (including the tenth) is equal to one.) That is:

And in general.

It’s the same story with zero: zero to any natural power is equal to zero. Therefore, .

As you can see, compared to square roots, it’s more difficult to figure out which number gives us the radical number to one degree or anothera . More difficult pick up answer and check it for correctness by raising it to a powern . The situation is greatly simplified if you know the powers of popular numbers in person. So now we are training. :) Let's recognize the degrees!)

Answers (in disarray):

Yes, yes! There are more answers than tasks.) Because, for example, 2 8, 4 4 and 16 2 are all the same number 256.

Have you practiced? Then let's look at some examples:

Answers (also in disarray): 6; 2; 3; 2; 3; 5.

Did it work? Fabulous! Let's move on.)

Limitations in the roots. Arithmetic rootnth degree.

The nth roots, like square roots, also have their own limitations and their own tricks. In essence, they are no different from those restrictions for square roots.

It doesn’t fit, right? What is 3, what is -3 to the fourth power will be +81. :) And with any root even degrees from a negative number will be the same song. And this means that It is impossible to extract roots of even degree from negative numbers . This is a taboo action in mathematics. It is as forbidden as dividing by zero. Therefore, expressions such as , and the like - don't make sense.

But the roots odd powers of negative numbers – please!

For example, ; , and so on.)

And from positive numbers you can extract any roots, of any degrees, with peace of mind:

In general, it’s understandable, I think.) And, by the way, the root does not have to be extracted exactly. These are just examples, purely for understanding.) It happens that in the process of solving (for example, equations) rather bad roots emerge. Something like . The cube root can be extracted perfectly from an eight, but here there is a seven under the root. What to do? It's OK. Everything is exactly the same.is a number that, when cubed, will give us 7. Only this number is very ugly and shaggy. Here it is:

Moreover, this number never ends and has no period: the numbers follow completely randomly. It is irrational... In such cases, the answer is left in the form of a root.) But if the root is extracted purely (for example, ), then, naturally, the root must be calculated and written down:

Again we take our experimental number 81 and extract the fourth root from it:

Because three in the fourth will be 81. Well, good! But also minus three in the fourth there will also be 81!

This results in ambiguity:

And, in order to eliminate it, just as in square roots, a special term was introduced: arithmetic rootnth degree from among a - this is what it is non-negative number,n-th degree of which is equal to a .

And the answer with plus or minus is called differently - algebraic rootnth degree. Any even power has an algebraic root two opposite numbers. At school they only work with arithmetic roots. Therefore, negative numbers in arithmetic roots are simply discarded. For example, they write: . The plus itself, of course, is not written: it imply.

Everything seems simple, but... But what about odd roots of negative numbers? After all, when you extract it, you always get a negative number! Since any negative number in odd degree also gives a negative number. And the arithmetic root only works with non-negative numbers! That's why it's arithmetic.)

In such roots, this is what they do: they take out the minus sign from under the root and place it in front of the root. Like this:

In such cases it is said that expressed through an arithmetic (i.e. already non-negative) root .

But there is one point that can cause confusion - this is the solution of simple equations with powers. For example, here's the equation:

We write the answer: . In fact, this answer is just a shorthand version of two answers:

The misunderstanding here is that I already wrote a little higher that at school only non-negative (i.e. arithmetic) roots are considered. And here is one of the answers with a minus... What should I do? No way! The signs here are result of solving the equation. A the root itself– the value is still non-negative! See for yourself:

Well, is it clearer now? With brackets?)

With an odd degree everything is much simpler - it always works out there one root. With a plus or a minus. For example:

So if we Just we extract the root (of even degree) from a number, then we always get one non-negative result. Because it is an arithmetic root. But if we decide equation with an even degree, then we get two opposite roots, since this is solution to the equation.

There are no problems with odd roots (cubic, fifth, etc.). Let’s take it out for ourselves and don’t worry about the signs. A plus under the root means the result of extraction is a plus. Minus means minus.)

And now it's time to meet properties of roots. Some will already be familiar to us from square roots, but several new ones will be added. Let's go!

Properties of roots. The root of the work.

This property is already familiar to us from square roots. For roots of other degrees everything is similar:

That is, the root of the product is equal to the product of the roots of each factor separately.

If the indicatorn even, then both radicalsa Andb must, naturally, be non-negative, otherwise the formula makes no sense. In the case of an odd exponent, there are no restrictions: we move the minuses forward from under the roots and then work with arithmetic roots.)

As with square roots, this formula is equally useful from left to right as from right to left. Applying the formula from left to right allows you to extract roots from the work. For example:

This formula, by the way, is valid not only for two, but for any number of factors. For example:

This formula can also be used to extract roots from large numbers: to do this, the number under the root is decomposed into smaller factors, and then the roots are extracted separately from each factor.

For example, this task:

The number is quite large. Is the root extracted from it? smooth– it’s also unclear without a calculator. It would be nice to factor it out. What exactly is the number 3375 divisible by? It looks like 5: the last digit is five.) Divide:

Oops, divisible by 5 again! 675:5 = 135. And 135 is again divisible by five. When will this end!)

135:5 = 27. With the number 27, everything is already clear - it’s three cubed. Means,

Then:

We extracted the root piece by piece, and that’s okay.)

Or this example:

Again we factorize according to the criteria of divisibility. Which one? At 4, because the last couple of digits 40 is divisible by 4. And by 10, because the last digit is zero. This means we can divide by 40 in one fell swoop:

We already know about the number 216 that it is a six cubed. Therefore,

And 40, in turn, can be expanded as . Then

And then we finally get:

It didn’t work out to extract the root cleanly, but that’s okay. Anyway, we simplified the expression: we know that under the root (even square, even cubic - any) it is customary to leave the smallest possible number.) In this example, we performed one very useful operation, also already familiar to us from square roots. Do you recognize? Yes! We carried out multipliers from the root. In this example, we took out a two and a six, i.e. number 12.

How to take the multiplier out of the root sign?

Taking a factor (or factors) beyond the root sign is very simple. We factor the radical expression and extract what is extracted.) And what is not extracted, we leave under the root. See:

We factor the number 9072. Since we have a fourth root, first of all we try to factorize it into factors that are fourth powers of natural numbers - 16, 81, etc.

Let's try to divide 9072 by 16:

Shared!

But 567 seems to be divisible by 81:

Means, .

Then

Properties of roots. Multiplying roots.

Let us now consider the reverse application of the formula - from right to left:

At first glance, nothing new, but appearances are deceiving.) Reverse application of the formula significantly expands our capabilities. For example:

Hmm, so what's wrong with that? They multiplied it and that's it. There's really nothing special here. Normal multiplication of roots. Here's an example!

The roots cannot be extracted purely from the factors separately. But the result is excellent.)

Again, the formula is valid for any number of factors. For example, you need to calculate the following expression:

The main thing here is attention. The example contains different roots – cube and fourth degree. And none of them are definitely extracted...

And the formula for the product of roots is applicable only to roots with identical indicators. Therefore, we will group cube roots into a separate group and fourth-degree roots into a separate group. And then, you see, everything will grow together.))

And you didn't need a calculator.)

How to enter a multiplier under the root sign?

The next useful thing is adding a number to the root. For example:

Is it possible to remove the triple inside the root? Elementary! If we turn three into root, then the formula for the product of roots will work. So, let's turn three into a root. Since we have a root of the fourth degree, we will also turn it into a root of the fourth degree.) Like this:

Then

A root, by the way, can be made from any non-negative number. Moreover, to the degree we want (everything depends on the specific example). This will be the nth root of this very number:

And now - attention! A source of very serious errors! It’s not for nothing that I said here about non-negative numbers. The arithmetic root only works with these. If we have a negative number somewhere in the task, then we either leave the minus just like that, in front of the root (if it is outside), or we get rid of the minus under the root, if it is inside. I remind you, if under the root even degree is a negative number, then the expression doesn't make sense.

For example, this task. Enter the multiplier under the root sign:

If we now bring to the root minus two, then we will be cruelly mistaken:

What's wrong here? And the fact is that the fourth power, due to its parity, happily “ate” this minus, as a result of which an obviously negative number turned into a positive one. And the correct solution looks like this:

In roots of odd degrees, although the minus is not “eaten up,” it is also better to leave it outside:

Here the odd root is cubic, and we have every right to push the minus under the root too. But in such examples it is preferable to also leave the minus outside and write the answer expressed through an arithmetic (non-negative) root, since the root, although it has the right to life, is not arithmetic.

So, with entering the number under the root, everything is also clear, I hope.) Let's move on to the next property.

Properties of roots. Root of a fraction. Root division.

This property also completely replicates that of square roots. Only now we extend it to roots of any degree:

The root of a fraction is equal to the root of the numerator divided by the root of the denominator.

If n is even, then the numbera must be non-negative, and the numberb – strictly positive (you cannot divide by zero). In the case of an odd indicator, the only limitation will be .

This property allows you to easily and quickly extract roots from fractions:

The idea is clear, I think. Instead of working with the entire fraction, we move on to working separately with the numerator and separately with the denominator.) If the fraction is a decimal or, horror of horrors, a mixed number, then we first move on to ordinary fractions:

Now let's see how this formula works from right to left. Here, too, very useful opportunities emerge. For example, this example:

The roots cannot be exactly extracted from the numerator and denominator, but from the entire fraction it is fine.) You can solve this example in another way - remove the factor from under the root in the numerator and then reduce it:

As you please. The answer will always be the same – the correct one. If you don't make mistakes along the way.)

So, we’ve sorted out the multiplication/division of roots. Let's go up to the next step and consider the third property - root to the power And root of the power .

Root to degree. Root of the degree.

How to raise a root to a power? For example, let's say we have a number. Can this number be raised to a power? In a cube, for example? Certainly! Multiply the root by itself three times, and - according to the formula for the product of roots:

Here is the root and degree as if mutually destroyed or compensated. Indeed, if we raise a number that, when raised into a cube, will give us a three, into this very cube, then what do we get? We'll get a three, of course! And this will be the case for any non-negative number. In general terms:

If the exponents and the root are different, then there are no problems either. If you know the properties of degrees.)

If the exponent is less than the exponent of the root, then we simply push the degree under the root:

In general it will be:

The idea is clear: we raise the radical expression to a power, and then simplify it, removing the factors from under the root, if possible. Ifn even thena must be non-negative. Why is understandable, I think.) And ifn odd, then there are no restrictions ona no longer available:

Let's deal now with root of the degree . That is, it is not the root itself that will be raised to a power, but radical expression. There is nothing complicated here either, but there is much more room for mistakes. Why? Because negative numbers come into play, which can cause confusion in the signs. For now, let's start with the roots of odd powers - they are much simpler.

Let us have the number 2. Can we cube it? Certainly!

Now let’s take the cube root back from the figure eight:

We started with a two and returned to a two.) No wonder: the cube was compensated for by the reverse operation - the extraction of the cube root.

Another example:

Everything is fine here too. The degree and the root compensated for each other. In general, for roots of odd powers we can write the following formula:

This formula is valid for any real numbera . Either positive or negative.

That is, an odd degree and the root of the same degree always compensate each other and a radical expression is obtained. :)

But with even to some extent this trick may no longer work. See for yourself:

Nothing special here yet. The fourth degree and the root of the fourth degree also balanced each other and the result was simply two, i.e. radical expression. And for anyone non-negative the numbers will be the same. Now let's just replace two in this root with minus two. That is, let’s calculate the following root:

The minus of the two was successfully “burned out” due to the fourth degree. And as a result of extracting the root (arithmetic!) we got positive number. It was minus two, now it’s plus two.) But if we had simply thoughtlessly “reduced” the degree and the root (the same!), we would have

Which is a grave mistake, yes.

Therefore for even exponent, the formula for the root of a degree looks like this:

Here we have added the modulus sign, which is unloved by many, but there is nothing scary about it: thanks to it, the formula also works for any real numbera. And the module simply cuts off the cons:

Only in roots of the nth degree did an additional distinction between even and odd degrees appear. Even degrees, as we see, are more capricious, yes.)

Now let’s consider a new useful and very interesting property, already characteristic specifically of roots of the nth degree: if the exponent of the root and the exponent of the radical expression are multiplied (divided) by the same natural number, then the value of the root will not change.

It’s somewhat reminiscent of the basic property of a fraction, isn’t it? In fractions, we can also multiply (divide) the numerator and denominator by the same number (except zero). In fact, this property of roots is also a consequence of the basic property of a fraction. When we meet degree with rational exponent, then everything will become clear. What, how and where.)

Direct application of this formula allows us to simplify absolutely any roots from any degrees. Including, if the exponents of the radical expression and the root itself different. For example, you need to simplify the following expression:

Let's do it simply. To begin with, we select the fourth power of the tenth under the root and - go ahead! How? According to the properties of degrees, of course! We take the multiplier out from under the root or work using the formula for the root of the power.

But let’s simplify it using just this property. To do this, let’s represent the four under the root as:

And now - the most interesting thing - mentally shorten the index under the root (two) with the index of the root (four)! And we get: