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 in order 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 generally speaking .

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

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.

Happened? 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 the 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. 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 the 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:

You can also use this formula 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. That is,

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 (whether square, even cubic, whatever) 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 power 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. And 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 the 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 as well. 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 (cannot be divided 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 wish. 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:

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 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 powers. 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:

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 of the nth roots 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 raise a root to a natural power, 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. Convert a mixed number to 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.

Definition
Power function with exponent p is the function f (x) = x p, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0 .

For natural values ​​of the exponent, the power function is the product of n numbers equal to x:
.
It is defined for all valid .

For positive rational values ​​of the exponent, the power function is the product of n roots of degree m of the number x:
.
For odd m, it is defined for all real x. For even m, the power function is defined for non-negative ones.

For negative , the power function is determined by the formula:
.
Therefore, it is not defined at the point.

For irrational values ​​of the exponent p, the power function is determined by the formula:
,
where a is an arbitrary positive number not equal to one: .
When , it is defined for .
When , the power function is defined for .

Continuity. A power function is continuous in its domain of definition.

Properties and formulas of power functions for x ≥ 0

Here we will consider the properties of the power function for non-negative values ​​of the argument x. As stated above, for certain values ​​of the exponent p, the power function is also defined for negative values ​​of x. In this case, its properties can be obtained from the properties of , using even or odd. These cases are discussed and illustrated in detail on the page "".

A power function, y = x p, with exponent p has the following properties:
(1.1) defined and continuous on the set
at ,
at ;
(1.2) has many meanings
at ,
at ;
(1.3) strictly increases with ,
strictly decreases as ;
(1.4) at ;
at ;
(1.5) ;
(1.5*) ;
(1.6) ;
(1.7) ;
(1.7*) ;
(1.8) ;
(1.9) .

Proof of properties is given on the page “Power function (proof of continuity and properties)”

Roots - definition, formulas, properties

Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ... - a natural number greater than one.

You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.

Square root of x is a root of degree 2: .

Cube root of x is a root of degree 3: .

Even degree

For even powers n = 2 m, the root is defined for x ≥ 0 . A formula that is often used is valid for both positive and negative x:
.
For square root:
.

The order in which the operations are performed is important here - that is, first the square is performed, resulting in a non-negative number, and then the root is taken from it (the square root can be taken from a non-negative number). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.

Odd degree

For odd powers, the root is defined for all x:
;
.

Properties and formulas of roots

The root of x is a power function:
.
When x ≥ 0 the following formulas apply:
;
;
, ;
.

These formulas can also be applied for negative values ​​of variables. You just need to make sure that the radical expression of even powers is not negative.

Private values

The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .

Example. Root of roots

Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.

y = x p for different values ​​of the exponent p.

Here are graphs of the function for non-negative values ​​of the argument x. Graphs of a power function defined for negative values ​​of x are given on the page “Power function, its properties and graphs"

Inverse function

The inverse of a power function with exponent p is a power function with exponent 1/p.

If, then.

Derivative of a power function

Derivative of nth order:
;

Deriving formulas > > >

Integral of a power function

P ≠ - 1 ;
.

Power series expansion

At - 1 < x < 1 the following decomposition takes place:

Expressions using complex numbers

Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:

Next, we take into account that the argument φ is not uniquely defined:
,

Let's consider the case when q = 0 , that is, the exponent is a real number, t = p. Then
.

If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.

If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function.

If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different values ​​of kp:
.
However, subsequent values ​​give values ​​that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions whose arguments differ by multiples of 2π, have equal values. Therefore, with a further increase in k, we obtain the same values ​​of z p as for k = k 0 = 0, 1, 2, ... n-1.

Thus, an exponential function with a rational exponent is multivalued and has n values ​​(branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began.

In particular, a root of degree n has n values. As an example, consider the nth root of a real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x, .
.
So, for a square root, n = 2 ,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.

• An arithmetic root of a natural power n>=2 of a non-negative number a is some non-negative number, when raised to the power n, the number a is obtained.

It can be proven that for any non-negative a and natural n, the equation x^n=a will have one single non-negative root. It is this root that is called the arithmetic root of the nth degree of the number a.

The arithmetic root of the nth degree of a number is denoted as follows: n√a. The number a in this case is called a radical expression.

An arithmetic root of the second degree is called a square root, and an arithmetic root of the third degree is called a cube root.

Basic properties of the arithmetic root of the nth degree

• 1. (n√a)^n = a.

For example, (5√2)^5 = 2.

This property follows directly from the definition of the nth arithmetic root.

If a is greater than or equal to zero, b is greater than zero and n, m are some natural numbers such that n is greater than or equal to 2 and m is greater than or equal to 2, then the following properties are true:

• 2. n√(a*b)= n√a*n√b.

For example, 4√27 * 4√3 = 4√(27*3) = 4√81 =4√(3^4) = 3.

• 3. n√(a/b) = (n√a)/(n√b).

For example, 3√(256/625) :3√(4/5) = 3√((256/625) : (4/5)) = (3√(64))/(3√(125)) = 4/5.

• 4. (n√a)^m = n√(a^m).

For example, 7√(5^21) = 7√((5^7)^3)) = (7√(5^7))^3 = 5^3 = 125.

• 5. m√(n√a) = (n*m) √a.

For example, 3√(4√4096) = 12√4096 = 12√(2^12) = 2.

Note that in property 2, the number b can be equal to zero, and in property 4, the number m can be any integer, provided that a>0.

Proof of the second property

All the last four properties can be proven in a similar way, so we will limit ourselves to proving only the second: n√(a*b)= n√a*n√b.

Using the definition of an arithmetic root, we prove that n√(a*b)= n√a*n√b.

To do this, we prove two facts: n√a*n√b. Greater than or equal to zero, and that (n√a*n√b.)^n = ab.

• 1. n√a*n√b is greater than or equal to zero, since both a and b are greater than or equal to zero.
• 2. (n√a*n√b)^n = a*b, since (n√a*n√b)^n = (n√a)^n *(n√b)^n = a* b.

Q.E.D. So the property is true. These properties will often have to be used when simplifying expressions containing arithmetic roots.

Congratulations: today we will look at roots - one of the most mind-blowing topics in 8th grade. :)

Many people get confused about roots, not because they are complex (what’s so complicated about it - a couple of definitions and a couple more properties), but because in most school textbooks roots are defined through such a jungle that only the authors of the textbooks themselves can understand this writing. And even then only with a bottle of good whiskey. :)

Therefore, now I will give the most correct and most competent definition of a root - the only one that you really should remember. And then I’ll explain: why all this is needed and how to apply it in practice.

But first, remember one important point that many textbook compilers for some reason “forget”:

Roots can be of even degree (our favorite $\sqrt(a)$, as well as all sorts of $\sqrt(a)$ and even $\sqrt(a)$) and odd degree (all sorts of $\sqrt(a)$, $\ sqrt(a)$, etc.). And the definition of a root of an odd degree is somewhat different from an even one.

Probably 95% of all errors and misunderstandings associated with roots are hidden in this fucking “somewhat different”. So let's clear up the terminology once and for all:

Definition. Even root n from the number $a$ is any non-negative the number $b$ is such that $((b)^(n))=a$. And the odd root of the same number $a$ is generally any number $b$ for which the same equality holds: $((b)^(n))=a$.

In any case, the root is denoted like this:

\(a)\]

The number $n$ in such a notation is called the root exponent, and the number $a$ is called the radical expression. In particular, for $n=2$ we get our “favorite” square root (by the way, this is a root of even degree), and for $n=3$ we get a cubic root (odd degree), which is also often found in problems and equations.

Examples. Classic examples of square roots:

\[\begin(align) & \sqrt(4)=2; \\ & \sqrt(81)=9; \\ & \sqrt(256)=16. \\ \end(align)\]

By the way, $\sqrt(0)=0$, and $\sqrt(1)=1$. This is quite logical, since $((0)^(2))=0$ and $((1)^(2))=1$.

Cube roots are also common - no need to be afraid of them:

\[\begin(align) & \sqrt(27)=3; \\ & \sqrt(-64)=-4; \\ & \sqrt(343)=7. \\ \end(align)\]

Well, a couple of “exotic examples”:

\[\begin(align) & \sqrt(81)=3; \\ & \sqrt(-32)=-2. \\ \end(align)\]

If you don’t understand what the difference is between an even and an odd degree, re-read the definition again. It is very important!

In the meantime, we will consider one unpleasant feature of roots, because of which we needed to introduce a separate definition for even and odd exponents.

Why are roots needed at all?

After reading the definition, many students will ask: “What were the mathematicians smoking when they came up with this?” And really: why are all these roots needed at all?

To answer this question, let's go back to elementary school for a moment. Remember: in those distant times, when the trees were greener and the dumplings tastier, our main concern was to multiply numbers correctly. Well, something like “five by five – twenty-five”, that’s all. But you can multiply numbers not in pairs, but in triplets, quadruples and generally whole sets:

\[\begin(align) & 5\cdot 5=25; \\ & 5\cdot 5\cdot 5=125; \\ & 5\cdot 5\cdot 5\cdot 5=625; \\ & 5\cdot 5\cdot 5\cdot 5\cdot 5=3125; \\ & 5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5=15\ 625. \end(align)\]

However, this is not the point. The trick is different: mathematicians are lazy people, so they had a hard time writing down the multiplication of ten fives like this:

That's why they came up with degrees. Why not write the number of factors as a superscript instead of a long string? Something like this:

It's very convenient! All calculations are reduced significantly, and you don’t have to waste a bunch of sheets of parchment and notebooks to write down some 5,183. This record was called a power of a number; a bunch of properties were found in it, but the happiness turned out to be short-lived.

After a grandiose drinking party, which was organized just for the “discovery” of degrees, some particularly stubborn mathematician suddenly asked: “What if we know the degree of a number, but the number itself is unknown?” Now, indeed, if we know that a certain number $b$, say, to the 5th power gives 243, then how can we guess what the number $b$ itself is equal to?

This problem turned out to be much more global than it might seem at first glance. Because it turned out that for most “ready-made” powers there are no such “initial” numbers. Judge for yourself:

\[\begin(align) & ((b)^(3))=27\Rightarrow b=3\cdot 3\cdot 3\Rightarrow b=3; \\ & ((b)^(3))=64\Rightarrow b=4\cdot 4\cdot 4\Rightarrow b=4. \\ \end(align)\]

What if $((b)^(3))=50$? It turns out that we need to find a certain number that, when multiplied by itself three times, will give us 50. But what is this number? It is clearly greater than 3, since 3 3 = 27< 50. С тем же успехом оно меньше 4, поскольку 4 3 = 64 >50. That is this number lies somewhere between three and four, but you won’t understand what it is equal to.

This is precisely why mathematicians came up with $n$th roots. This is precisely why the radical symbol $\sqrt(*)$ was introduced. To designate the very number $b$, which to the indicated degree will give us a previously known value

\[\sqrt[n](a)=b\Rightarrow ((b)^(n))=a\]

I don’t argue: often these roots are easily calculated - we saw several such examples above. But still, in most cases, if you think of an arbitrary number and then try to extract the root of an arbitrary degree from it, you will be in for a terrible bummer.

What is there! Even the simplest and most familiar $\sqrt(2)$ cannot be represented in our usual form - as an integer or a fraction. And if you enter this number into a calculator, you will see this:

\[\sqrt(2)=1.414213562...\]

As you can see, after the decimal point there is an endless sequence of numbers that do not obey any logic. You can, of course, round this number to quickly compare with other numbers. For example:

\[\sqrt(2)=1.4142...\approx 1.4 \lt 1.5\]

Or here's another example:

\[\sqrt(3)=1.73205...\approx 1.7 \gt 1.5\]

But all these roundings, firstly, are quite rough; and secondly, you also need to be able to work with approximate values, otherwise you can catch a bunch of non-obvious errors (by the way, the skill of comparison and rounding is required to be tested on the profile Unified State Examination).

Therefore, in serious mathematics you cannot do without roots - they are the same equal representatives of the set of all real numbers $\mathbb(R)$, just like the fractions and integers that have long been familiar to us.

The inability to represent a root as a fraction of the form $\frac(p)(q)$ means that this root is not a rational number. Such numbers are called irrational, and they cannot be accurately represented except with the help of a radical or other constructions specially designed for this (logarithms, powers, limits, etc.). But more on that another time.

Let's consider several examples where, after all the calculations, irrational numbers will still remain in the answer.

\[\begin(align) & \sqrt(2+\sqrt(27))=\sqrt(2+3)=\sqrt(5)\approx 2.236... \\ & \sqrt(\sqrt(-32 ))=\sqrt(-2)\approx -1.2599... \\ \end(align)\]

Naturally, from the appearance of the root it is almost impossible to guess what numbers will come after the decimal point. However, you can count on a calculator, but even the most advanced date calculator only gives us the first few digits of an irrational number. Therefore, it is much more correct to write the answers in the form $\sqrt(5)$ and $\sqrt(-2)$.

This is exactly why they were invented. To conveniently record answers.

Why are two definitions needed?

The attentive reader has probably already noticed that all the square roots given in the examples are taken from positive numbers. Well, at least from scratch. But cube roots can be calmly extracted from absolutely any number - be it positive or negative.

Why is this happening? Take a look at the graph of the function $y=((x)^(2))$:

The graph of a quadratic function gives two roots: positive and negative

Let's try to calculate $\sqrt(4)$ using this graph. To do this, a horizontal line $y=4$ is drawn on the graph (marked in red), which intersects with the parabola at two points: $((x)_(1))=2$ and $((x)_(2)) =-2$. This is quite logical, since

Everything is clear with the first number - it is positive, so it is the root:

But then what to do with the second point? Like four has two roots at once? After all, if we square the number −2, we also get 4. Why not write $\sqrt(4)=-2$ then? And why do teachers look at such posts as if they want to eat you? :)

The trouble is that if you don’t impose any additional conditions, then the quad will have two square roots - positive and negative. And any positive number will also have two of them. But negative numbers will have no roots at all - this can be seen from the same graph, since the parabola never falls below the axis y, i.e. does not accept negative values.

A similar problem occurs for all roots with an even exponent:

1. Strictly speaking, each positive number will have two roots with even exponent $n$;
2. From negative numbers, the root with even $n$ is not extracted at all.

That is why in the definition of a root of an even degree $n$ it is specifically stipulated that the answer must be a non-negative number. This is how we get rid of ambiguity.

But for odd $n$ there is no such problem. To see this, let's look at the graph of the function $y=((x)^(3))$:

A cube parabola can take any value, so the cube root can be taken from any number

Two conclusions can be drawn from this graph:

1. The branches of a cubic parabola, unlike a regular one, go to infinity in both directions - both up and down. Therefore, no matter what height we draw a horizontal line, this line will certainly intersect with our graph. Consequently, the cube root can always be extracted from absolutely any number;
2. In addition, such an intersection will always be unique, so you don’t need to think about which number is considered the “correct” root and which one to ignore. That is why determining roots for an odd degree is simpler than for an even degree (there is no requirement for non-negativity).

It's a pity that these simple things are not explained in most textbooks. Instead, our brains begin to soar with all sorts of arithmetic roots and their properties.

Yes, I don’t argue: you also need to know what an arithmetic root is. And I will talk about this in detail in a separate lesson. Today we will also talk about it, because without it all thoughts about roots of $n$-th multiplicity would be incomplete.

But first you need to clearly understand the definition that I gave above. Otherwise, due to the abundance of terms, such a mess will begin in your head that in the end you will not understand anything at all.

All you need to do is understand the difference between even and odd indicators. Therefore, let’s once again collect everything you really need to know about roots:

1. A root of an even degree exists only from a non-negative number and is itself always a non-negative number. For negative numbers such a root is undefined.
2. But the root of an odd degree exists from any number and can itself be any number: for positive numbers it is positive, and for negative numbers, as the cap hints, it is negative.

Is it difficult? No, it's not difficult. It's clear? Yes, it’s completely obvious! So now we will practice a little with the calculations.

Basic properties and limitations

Roots have many strange properties and limitations - this will be discussed in a separate lesson. Therefore, now we will consider only the most important “trick”, which applies only to roots with an even index. Let's write this property as a formula:

\[\sqrt(((x)^(2n)))=\left| x\right|\]

In other words, if we raise a number to an even power and then extract the root of the same power, we will not get the original number, but its modulus. This is a simple theorem that can be easily proven (it is enough to consider non-negative $x$ separately, and then negative ones separately). Teachers constantly talk about it, it is given in every school textbook. But as soon as it comes to solving irrational equations (i.e., equations containing a radical sign), students unanimously forget this formula.

To understand the issue in detail, let's forget all the formulas for a minute and try to calculate two numbers straight ahead:

\[\sqrt(((3)^(4)))=?\quad \sqrt(((\left(-3 \right))^(4)))=?\]

These are very simple examples. Most people will solve the first example, but many people get stuck on the second. To solve any such crap without problems, always consider the procedure:

1. First, the number is raised to the fourth power. Well, it's kind of easy. You will get a new number that can be found even in the multiplication table;
2. And now from this new number it is necessary to extract the fourth root. Those. no “reduction” of roots and powers occurs - these are sequential actions.

Let's look at the first expression: $\sqrt(((3)^(4)))$. Obviously, you first need to calculate the expression under the root:

\[((3)^(4))=3\cdot 3\cdot 3\cdot 3=81\]

Then we extract the fourth root of the number 81:

Now let's do the same with the second expression. First, we raise the number −3 to the fourth power, which requires multiplying it by itself 4 times:

\[((\left(-3 \right))^(4))=\left(-3 \right)\cdot \left(-3 \right)\cdot \left(-3 \right)\cdot \ left(-3 \right)=81\]

We got a positive number, since the total number of minuses in the product is 4, and they will all cancel each other out (after all, a minus for a minus gives a plus). Then we extract the root again:

In principle, this line could not have been written, since it’s a no brainer that the answer would be the same. Those. an even root of the same even power “burns” the minuses, and in this sense the result is indistinguishable from a regular module:

\[\begin(align) & \sqrt(((3)^(4)))=\left| 3 \right|=3; \\ & \sqrt(((\left(-3 \right))^(4)))=\left| -3 \right|=3. \\ \end(align)\]

These calculations are in good agreement with the definition of a root of an even degree: the result is always non-negative, and the radical sign also always contains a non-negative number. Otherwise, the root is undefined.

Note on procedure

1. The notation $\sqrt(((a)^(2)))$ means that we first square the number $a$ and then take the square root of the resulting value. Therefore, we can be sure that there is always a non-negative number under the root sign, since $((a)^(2))\ge 0$ in any case;
2. But the notation $((\left(\sqrt(a) \right))^(2))$, on the contrary, means that we first take the root of a certain number $a$ and only then square the result. Therefore, the number $a$ can in no case be negative - this is a mandatory requirement included in the definition.

Thus, in no case should one thoughtlessly reduce roots and degrees, thereby allegedly “simplifying” the original expression. Because if the root has a negative number and its exponent is even, we get a bunch of problems.

However, all these problems are relevant only for even indicators.

Removing the minus sign from under the root sign

Naturally, roots with odd exponents also have their own feature, which in principle does not exist with even ones. Namely:

\[\sqrt(-a)=-\sqrt(a)\]

In short, you can remove the minus from under the sign of roots of odd degrees. This is a very useful property that allows you to “throw out” all the disadvantages:

\[\begin(align) & \sqrt(-8)=-\sqrt(8)=-2; \\ & \sqrt(-27)\cdot \sqrt(-32)=-\sqrt(27)\cdot \left(-\sqrt(32) \right)= \\ & =\sqrt(27)\cdot \sqrt(32)= \\ & =3\cdot 2=6. \end(align)\]

This simple property greatly simplifies many calculations. Now you don’t need to worry: what if a negative expression was hidden under the root, but the degree at the root turned out to be even? It is enough just to “throw out” all the minuses outside the roots, after which they can be multiplied by each other, divided, and generally do many suspicious things, which in the case of “classical” roots are guaranteed to lead us to an error.

And here another definition comes onto the scene - the same one with which in most schools they begin the study of irrational expressions. And without which our reasoning would be incomplete. Meet!

Arithmetic root

Let's assume for a moment that under the root sign there can only be positive numbers or, in extreme cases, zero. Let's forget about even/odd indicators, let's forget about all the definitions given above - we will work only with non-negative numbers. What then?

And then we will get an arithmetic root - it partially overlaps with our “standard” definitions, but still differs from them.

Definition. An arithmetic root of the $n$th degree of a non-negative number $a$ is a non-negative number $b$ such that $((b)^(n))=a$.

As we can see, we are no longer interested in parity. Instead, a new restriction appeared: the radical expression is now always non-negative, and the root itself is also non-negative.

To better understand how the arithmetic root differs from the usual one, take a look at the graphs of the square and cubic parabola we are already familiar with:

Arithmetic root search area - non-negative numbers

As you can see, from now on we are only interested in those pieces of graphs that are located in the first coordinate quarter - where the coordinates $x$ and $y$ are positive (or at least zero). You no longer need to look at the indicator to understand whether we have the right to put a negative number under the root or not. Because negative numbers are no longer considered in principle.

You may ask: “Well, why do we need such a neutered definition?” Or: “Why can’t we get by with the standard definition given above?”

Well, I will give just one property because of which the new definition becomes appropriate. For example, the rule for exponentiation:

\[\sqrt[n](a)=\sqrt(((a)^(k)))\]

Please note: we can raise the radical expression to any power and at the same time multiply the root exponent by the same power - and the result will be the same number! Here are examples:

\[\begin(align) & \sqrt(5)=\sqrt(((5)^(2)))=\sqrt(25) \\ & \sqrt(2)=\sqrt(((2)^ (4)))=\sqrt(16)\\ \end(align)\]

So what's the big deal? Why couldn't we do this before? Here's why. Let's consider a simple expression: $\sqrt(-2)$ - this number is quite normal in our classical understanding, but absolutely unacceptable from the point of view of the arithmetic root. Let's try to convert it:

$\begin(align) & \sqrt(-2)=-\sqrt(2)=-\sqrt(((2)^(2)))=-\sqrt(4) \lt 0; \\ & \sqrt(-2)=\sqrt(((\left(-2 \right))^(2)))=\sqrt(4) \gt 0. \\ \end(align)$

As you can see, in the first case we removed the minus from under the radical (we have every right, since the exponent is odd), and in the second case we used the above formula. Those. From a mathematical point of view, everything is done according to the rules.

WTF?! How can the same number be both positive and negative? No way. It’s just that the formula for exponentiation, which works great for positive numbers and zero, begins to produce complete heresy in the case of negative numbers.

It was in order to get rid of such ambiguity that arithmetic roots were invented. A separate large lesson is devoted to them, where we consider all their properties in detail. So we won’t dwell on them now - the lesson has already turned out to be too long.

Algebraic root: for those who want to know more

I thought for a long time whether to put this topic in a separate paragraph or not. In the end I decided to leave it here. This material is intended for those who want to understand the roots even better - no longer at the average “school” level, but at one close to the Olympiad level.

So: in addition to the “classical” definition of the $n$th root of a number and the associated division into even and odd exponents, there is a more “adult” definition that does not depend at all on parity and other subtleties. This is called an algebraic root.

Definition. The algebraic $n$th root of any $a$ is the set of all numbers $b$ such that $((b)^(n))=a$. There is no established designation for such roots, so we’ll just put a dash on top:

\[\overline(\sqrt[n](a))=\left\( b\left| b\in \mathbb(R);((b)^(n))=a \right. \right\) \]

The fundamental difference from the standard definition given at the beginning of the lesson is that an algebraic root is not a specific number, but a set. And since we work with real numbers, this set comes in only three types:

1. Empty set. Occurs when you need to find an algebraic root of an even degree from a negative number;
2. A set consisting of one single element. All roots of odd powers, as well as roots of even powers of zero, fall into this category;
3. Finally, the set can include two numbers - the same $((x)_(1))$ and $((x)_(2))=-((x)_(1))$ that we saw on the graph quadratic function. Accordingly, such an arrangement is possible only when extracting the root of an even degree from a positive number.

The last case deserves more detailed consideration. Let's count a couple of examples to understand the difference.

Example. Evaluate the expressions:

\[\overline(\sqrt(4));\quad \overline(\sqrt(-27));\quad \overline(\sqrt(-16)).\]

Solution. The first expression is simple:

\[\overline(\sqrt(4))=\left\( 2;-2 \right\)\]

It is two numbers that are part of the set. Because each of them squared gives a four.

\[\overline(\sqrt(-27))=\left\( -3 \right\)\]

Here we see a set consisting of only one number. This is quite logical, since the root exponent is odd.

Finally, the last expression:

\[\overline(\sqrt(-16))=\varnothing \]

We received an empty set. Because there is not a single real number that, when raised to the fourth (i.e., even!) power, will give us the negative number −16.

Final note. Please note: it was not by chance that I noted everywhere that we work with real numbers. Because there are also complex numbers - it is quite possible to calculate $\sqrt(-16)$ there, and many other strange things.

However, complex numbers almost never appear in modern school mathematics courses. They have been removed from most textbooks because our officials consider the topic “too difficult to understand.”

That's all. In the next lesson we will look at all the key properties of roots and finally learn how to simplify irrational expressions. :)