The root is equal to the expression. Square root. Detailed theory with examples. Algebraic root: for those who want to know more

In this article, we will introduce root concept... We will proceed sequentially: we will start with the square root, from it we will move on to the description of the cube root, after that we will generalize the concept of the root by defining the n-th root. At the same time, we will introduce definitions, designations, give examples of roots and give the necessary explanations and comments.

Square root, arithmetic square root

To understand the definition of root of a number, and square root in particular, you need to have. At this point, we will often come across the second power of a number - the square of a number.

Let's start with definition of square root.

Definition

Square root of a is a number whose square is a.

In order to bring examples of square roots, we 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, according to the above definition, 5 is the square root of 25, −0.3 and 0.3 are square roots of 0.09, and 0 is the square root of zero.

It should be noted that not for every number a exists whose square is equal to a. Namely, for any negative number a, there is not a single real number b whose square would be equal to a. Indeed, the equality a = b 2 is impossible for any negative a, since b 2 is a non-negative number for any b. In this way, 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 does not make sense.

This leads to a logical question: "Is there a square root of a for any non-negative a"? The answer is yes. The rationale for this fact can be considered a constructive method used to find the value of the square root.

Then the following logical question arises: "What is the number of all square roots from 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 from the number a is equal to two, and the roots are. Let us 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 us prove that 0 is the only square root of zero. Let's use the method by contradiction. 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 nonzero b the value of the expression b 2 is positive. We have come to a contradiction. This proves that 0 is the only square root of zero.

We pass to the cases when a is a positive number. Above we said that there is always a square root of any non-negative number, let the square root of a be the number b. Suppose there is a number c, which is also the square root of a. Then, by the definition of the square root, the equalities b 2 = a and c 2 = a hold, 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 due to properties of actions with real numbers is possible only if 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, with square roots being opposite numbers.

For the convenience of working with square roots, the negative root is "separated" from the positive one. For this purpose, arithmetic square root definition.

Definition

Arithmetic square root of a non-negative number a Is a non-negative number whose square is a.

The notation is adopted for the arithmetic square root of the number a. The sign is called the arithmetic square root sign. It is also called the radical sign. Therefore, you can partly hear both "root" and "radical", which means the same object.

The number under the sign of the arithmetic square root is called root 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 record the number 151 is a radical number, and in the record the expression a is a radical expression.

When reading the word "arithmetic" is often omitted, for example, the record is read as "the square root of seven point twenty-nine hundredths." The word "arithmetic" is pronounced only when they want to emphasize that we are talking about the positive square root of a number.

In the light of the introduced notation, it follows from the definition of the arithmetic square root that for any non-negative number a.

Square roots of a positive number a are written as and using the arithmetic square root sign. 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 make sense of the notation until we study complex numbers... For example, the expressions and are meaningless.

On the basis of the definition of the square root, the properties of square roots are proved, which are often used in practice.

In conclusion of this item, note that the square roots of the number a are solutions of the form x 2 = a with respect to the variable x.

Cubic root of a number

Determining the cube root of the number a is given similarly to the definition of a square root. Only it is based on the concept of a cube of a number, not a square.

Definition

Cubic root of number a is a number whose cube is equal to a.

Let us 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 the cube root, we can argue 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 the number a, in contrast to the square root, always exists, and 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 the square root.

Moreover, there is only a single cube root of a given number a. Let us prove the last statement. For this, we will separately consider three cases: a is a positive number, a = 0 and a is a negative number.

It is easy to show that for a positive a, the cube root of a cannot be negative or zero. Indeed, let b be a 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 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 besides the number b there is one more cube root of the number a, we denote it by 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 obtained 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-hand 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 cubic root of a positive number a.

For a = 0, only the number zero is the cube root of the number a. Indeed, if we assume that there is a number b, which is a nonzero cube root of zero, then the equality b 3 = 0 must hold, which is possible only when b = 0.

For negative a, one can argue similar to the case for positive a. First, we show that the cube root of a negative number cannot be equal to either a positive number or zero. Second, 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 the only one.

Let's give arithmetic cube root definition.

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 exponent... The number under the root sign is root number, the expression under the root sign is root expression.

Although the arithmetic cube root is defined only for non-negative numbers a, it is also convenient to use notations in which negative numbers are under the sign of the arithmetic cube root. We will understand them as follows:, where a is a positive number. For instance, .

We will talk about the properties of cube roots in the general article on properties of roots.

Calculation of the cube root value is called cube root extraction, this action is discussed in the article root extraction: methods, examples, solutions.

In conclusion of this paragraph, we say that the cube root of the number a is a solution of the form x 3 = a.

Nth root, nth arithmetic root

To generalize the concept of a root of a number, we introduce determining the root of the nth degree for n.

Definition

Nth root of a Is a number whose n -th power is a.

From this definition, it is clear that the root of the first degree 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 considered special cases of the nth root for n = 2 and n = 3 - square root and cube root. That is, the square root is the root of the second degree, and the cube root is the root of the third degree. To study the roots of the n-th 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, for n = 5, 7, 9, ...). This is due to the fact that roots of even degrees are analogous to a square root, and roots of odd degrees are analogous to a cubic root. Let's deal with them in turn.

Let's start with the roots, whose powers are even numbers 4, 6, 8, ... As we said, they are analogous to the square root of the number a. That is, the root of any even degree from the number a exists only for a 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 an even degree from the number a, and they are opposite numbers.

Let us substantiate the last statement. Let b be a root of an even degree (we denote it as 2 m, where m is some natural number) from the number a. Suppose there is a number c - one more root of degree 2 m of the number a. Then b 2 m - c 2 m = a - a = 0. But we know of 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... This equality implies 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 cube root. That is, the root of any odd degree from the number a exists for any real number a, and for a given number a it is unique.

The uniqueness of a root of odd degree 2 m + 1 of a is proved by analogy with the proof of the uniqueness of a cubic 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 = (b − c) (b 2 m + b 2 m − 1 c + b 2 m − 2 c 2 +… + c 2 m)... The expression in the last parenthesis 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, for m = 2 we have b 5 −c 5 = (b − c) (b 4 + b 3 c + b 2 c 2 + b c 3 + c 4) = (b − c) (b 4 + c 4 + b c (b 2 + c 2 + b c))... When a and b are both positive or both negative, their product is a positive number, then the expression b 2 + c 2 + b · c in the highest nesting parentheses is positive as the sum of positive numbers. Now, moving sequentially to the expressions in parentheses of the previous degrees of nesting, we make sure 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 is possible only when b - c = 0, that is, when the number b is equal to the number c.

It's time to deal with the notation of the roots of the n-th degree. For this, it is given definition of the nth arithmetic root.

Definition

An arithmetic root of the nth degree of a non-negative number a is a non-negative number, the n -th power of which is equal to a.

The nth arithmetic root of a non-negative number a is denoted as. The number a is called the root number, and the number n is called the root indicator. For example, consider a record, here the radical number is 125.36, and the root exponent is 5.

Note that for n = 2 we are dealing with the square root of a number, in this case it is customary not to write the root exponent, that is, the records and mean the same number.

Despite the fact that the definition of the arithmetic root of the nth degree, as well as its designation, were introduced for non-negative radical numbers, for convenience, for odd root exponents and negative radical numbers, we will use notations of the form, which we will understand as. For instance, and .

We will not attach any meaning to roots of even degrees with negative radical numbers (until we start studying complex numbers). For example, expressions and do not make sense.

On the basis of the definition given above, the properties of roots of the n-th degree are substantiated, which have wide practical application.

In conclusion, it should be said that the roots of the n-th degree are the roots of equations of the form x n = a.

Practical results

The first practically important result: .

This result essentially reflects the definition of an even root. The ⇔ sign means equivalence. That is, the given record should be understood as follows: if, then, and if, then. And now the same thing, but in words: if b is a root of an even degree 2 k of the number a, then b is a non-negative number that satisfies the equality b 2 k = a, and vice versa, if b is a non-negative number that satisfies the equality b 2 k = a, that is, b is an even root of 2 k of a.

From the first equality of the system, it is clear that the number a is non-negative, since it is equal to the non-negative number b raised to an even power 2 · k.

Thus, the school considers the roots of even degrees only from non-negative numbers, they understand them as , and roots of even powers of negative numbers are not given any meaning.

The second practically important result: .

It essentially combines the definition of an arithmetic root of an odd power and the definition of an odd root of a negative number. Let us explain this.

From the definitions given in the previous paragraphs, it is clear that they give meaning to the roots of odd degrees from any real numbers, not only non-negative, but also negative. For non-negative numbers b, it is assumed that ... The last system implies the condition a≥0. For negative numbers −a (where a is a positive number), take ... It is clear that with this definition, it is a negative number, since it is equal, but there is a positive number. It is also clear that raising the root to the 2 · k + 1 power gives the root number –a. Indeed, taking into account this definition and properties of the degrees, we have

From this we conclude that the root of an odd degree 2 k + 1 of a negative number −a is a negative number b, the degree of 2 k + 1 of which is equal to −a, in literal form ... Combining results for a≥0 and for –a<0 , приходим к следующему выводу: корень нечетной степени 2·k+1 из произвольного действительного числа a есть число b (оно может быть как неотрицательным, так и отрицательным), которое при возведении в степень 2·k+1 равно a , то есть .

Thus, the school considers the roots of odd degrees from any real numbers and understands them as follows: .

In conclusion, we will once again write down the two results of interest to us: and .

Congratulations: today we will be examining roots - one of the most brain-bearing topics of the 8th grade. :)

Many are confused about the roots, not because they are complex (which is so difficult - a couple of definitions and a couple of properties), but because in most school textbooks the roots are determined through such a jungle that only the authors of the textbooks themselves can figure out this scribble. And even then only with a bottle of good whiskey. :)

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

But first, remember one important point, which for some reason many textbook compilers "forget":

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

Here in this fucking "somewhat different" hidden, probably 95% of all errors and misunderstandings associated with the roots. Therefore, let's deal with the terminology once and for all:

Definition. Even root n from $ a $ is any non-negative a number $ b $ 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 indicated like this:

\ (a) \]

The number $ n $ in such a record is called the exponent of the root, 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 an even root), and for $ n = 3 $ - cubic (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 $.

Cubic roots are also common - don't be afraid of them:

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

Well, and a couple of "exotic examples":

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

If you do not understand what is the difference between an even and an odd degree, read the definition again. It is very important!

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

Why do we need roots at all?

After reading the definition, many students will ask, "What did the mathematicians smoke when they came up with this?" Indeed: why do we need all these roots at all?

To answer this question, let's go back to elementary classes for a minute. Remember: in those distant times, when trees were greener and dumplings tastier, our main concern was to multiply numbers correctly. Well, something like "five by five - twenty five", that's all. But after all, you can multiply numbers not in pairs, but in triples, fours and, in general, 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 to write down the multiplication of ten fives like this:

So they came up with degrees. Why not superscript the number of factors instead of a long string? Like this:

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

After a huge booze, which was organized just about the "discovery" of degrees, some particularly stubborn mathematician suddenly asked: "What if we know the degree of a number, but we do not know the number itself?" Now, really, if we know that a certain number $ b $, for example, in the 5th power gives 243, then how can we guess what the number $ b $ 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 the majority of "ready" degrees 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 you need to find a certain number, which, being multiplied three times by itself, 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 what it is equal to - figs you will understand.

It is for this that mathematicians invented the roots of the $ n $ -th degree. This is why the radical symbol $ \ sqrt (*) $ was introduced. To designate the very number $ b $, which, to the specified degree, will give us a previously known value

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

I do not argue: these roots are often easily counted - we have seen several such examples above. Still, in most cases, if you guess an arbitrary number, and then try to extract an arbitrary root from it, you are in for a cruel 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 type this number into a calculator, you will see this:

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

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

\ [\ 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 rather rough; and secondly, you also need to be able to work with approximate values, otherwise you can catch a bunch of unobvious errors (by the way, the skill of comparison and rounding is mandatory checked on the profile exam).

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

The impossibility of representing 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 otherwise than with the help of a radical, or other specially designed constructions (logarithms, degrees, limits, etc.). But more about that another time.

Consider a few 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, by 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 perfect date calculator gives us only the first few digits of an irrational number. Therefore, it is much more correct to write the answers in the form of $ \ sqrt (5) $ and $ \ sqrt (-2) $.

That's why they were invented. To conveniently record the answers.

Why are two definitions needed?

The attentive reader has probably already noticed that all the square roots given in the examples are derived from positive numbers. Well, as a last resort from scratch. But the cube roots are 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 plot 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 chart (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, therefore it is the root:

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

The trouble is that if no additional conditions are imposed, then the four will have two square roots - positive and negative. And any positive number will also have two. 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:

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

That is why in the definition of the root of an even power of $ n $ it is specially 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 verify this, let's take a look at the graph of the function $ y = ((x) ^ (3)) $:

A cubic parabola takes any value, so the cube root is extracted from any number

Two conclusions can be drawn from this graph:

The branches of a cubic parabola, in contrast to the usual one, go to infinity in both directions - both up and down. Therefore, at whatever height we draw a horizontal line, this line will necessarily intersect with our graph. Consequently, the cube root can always be extracted from absolutely any number;
In addition, such an intersection will always be the only one, so there is no need to think about which number to consider the "correct" root, and which number to score. That is why the definition of roots for an odd degree is simpler than for an even one (there is no requirement of non-negativity).

It is a shame that these simple things are not explained in most textbooks. Instead, the brain begins to float to us with all sorts of arithmetic roots and their properties.

Yes, I do not argue: what is an arithmetic root - you also need to know. And I will cover this in detail in a separate tutorial. Today we will also talk about it, because without it all thoughts about the 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. So once again, let's put together everything you really need to know about roots:

An even root exists only from a non-negative number and is itself always a non-negative number. For negative numbers, such a root is undefined.

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 ones, as the cap hints at, negative.

Is it difficult? No, not difficult. Clear? Yes, in general, it is obvious! So now we are going to practice some calculations.

Basic properties and limitations

Roots have many strange properties and limitations - there will be a separate lesson about this. Therefore, now we will consider only the most important "trick", which applies only to roots with an even exponent. Let's write this property in the form of a formula:

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

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

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

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

These are very simple examples. The first example will be solved by most people, but on the second, many will stick. To solve any such crap without problems, always consider the order of actions:

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

We work with 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 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, for which we need to multiply 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 work is 4 pieces, and they will all be mutually destroyed (after all, minus by minus gives a plus). Then we extract the root again:

In principle, this line could not have been written, since it is a no-brainer that the answer will be the same. Those. an even root of the same even power “burns out” the minuses, and in this sense the result is indistinguishable from the usual modulus:

\ [\ 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 an even root: the result is always non-negative, and under the radical sign there is always a non-negative number. Otherwise, the root is undefined.

Procedure note

The notation $ \ sqrt (((a) ^ (2))) $ means that we first square the number $ a $ and then extract the square root from the resulting value. Therefore, we can be sure that a non-negative number always sits under the root sign, since $ ((a) ^ (2)) \ ge 0 $ in any case;
But the record $ ((\ left (\ sqrt (a) \ right)) ^ (2)) $, on the contrary, means that we first extract the root from 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 in the definition.

Thus, in no case should you mindlessly reduce the roots and degrees, thereby supposedly "simplifying" the original expression. Because if there is a negative number under the root, and its exponent is even, we get a bunch of problems.

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

Removing the minus from the root sign

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

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

In short, you can take out the minus from under the sign of the roots of an odd degree. This is a very useful property that allows you to "throw out" all the minuses out:

\ [\ 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 there is no need to worry: suddenly a negative expression has crept under the root, and the degree at the root turns 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 a mistake.

And here another definition comes into play - the very one with which in most schools the study of irrational expressions begins. And without which our reasoning would be incomplete. Please welcome!

Arithmetic root

Let's assume for a moment that there can be only positive numbers under the root sign, or at most 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 get the 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 you can see, we are no longer interested in parity. Instead, a new restriction has 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 already familiar square and cubic parabola graphs:

Arithmetic root search area - non-negative numbers

As you can see, from now on we are only interested in those parts of the 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 root a negative number or not. Because negative numbers are no longer considered in principle.

You may ask: "Well, why do we need such a castrated definition?" Or: "Why can't you 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 is:

\ [\ 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 some 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 have done this earlier? Here's why. Consider a simple expression: $ \ sqrt (-2) $ - this number is quite normal in our classical sense, but absolutely unacceptable from the point of view of the arithmetic root. Let's try to transform 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 indicator is odd), and in the second, we used the above formula. Those. from the point of view of mathematics, 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 exponentiation formula, which works great for positive numbers and zero, starts to be heresy when it comes to negative numbers.

In order to get rid of such ambiguity, they came up with arithmetic roots. A separate big lesson is devoted to them, where we consider in detail all their properties. So now we will not dwell on them - 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 here. This material is intended for those who want to understand the roots even better - not at an average "school" level, but at a level 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 indicators, there is a more "adult" definition that does not depend on parity and other subtleties at all. This is called an algebraic root.

Definition. The algebraic root of the $ n $ th degree of any $ a $ is the set of all numbers $ b $ such that $ ((b) ^ (n)) = a $. There is no well-established designation for such roots, so we 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, there are only three types of this set:

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

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

Example. Evaluate 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 make up the set. Because each of them in the square 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 got an empty set. Because there is not a single real number, which when raised to the fourth (i.e. even!) Degree will give us a negative number -16.

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

However, in the modern school mathematics course, complex numbers are almost never found. They were deleted from most textbooks because our officials consider this 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. :)

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In mathematics..1) a root of power n of a number a is any number x (denoted by, a is called a radical expression), the nth power of which is equal to a (). The action of finding the root is called root extraction2)] The root of the equation is a number that after ... ...

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Books

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I looked again at the sign ... And let's go!

Let's start with a simple one:

Just a minute. this, which means that we can write like this:

Got it? Here's the next one for you:

The roots of the resulting numbers are not exactly extracted? It doesn't matter - here are some examples:

But what if the factors are not two, but more? The same! The root multiplication formula works with any number of factors:

Now completely on its own:

Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!

Division of roots

We figured out the multiplication of roots, now we will proceed to the property of division.

Let me remind you that the general formula looks like this:

This means that the root of the quotient is equal to the quotient of the roots.

Well, let's figure it out with examples:

That's all science. Here's an example:

Everything is not as smooth as in the first example, but, as you can see, there is nothing complicated.

But what if an expression like this comes across:

You just need to apply the formula in the opposite direction:

And here's an example:

You can also come across this expression:

Everything is the same, only here you need to remember how to translate fractions (if you don't remember, look into the topic and come back!). Remembered? Now we decide!

I am sure that you have coped with everything, everything, now let's try to build roots in power.

Exponentiation

What happens if the square root is squared? It's simple, let's remember the meaning of the square root of a number - this is a number whose square root is equal to.

So, if we raise a number whose square root is equal to the square, then what do we get?

Well, of course, !

Let's look at examples:

It's simple, right? And if the root is in a different degree? Nothing wrong!

Stick to the same logic and remember the properties and possible actions with degrees.

Read the theory on the topic "" and everything will become very clear to you.

For example, here's an expression:

In this example, the degree is even, but what if it is odd? Again, apply the power properties and factor everything:

With this, everything seems to be clear, but how to extract the root of a number to a power? For example, this is:

Pretty simple, right? And if the degree is more than two? We follow the same logic using degree properties:

Well, is everything clear? Then solve the examples yourself:

And here are the answers:

Introduction under the root sign

What have we not learned to do with roots! It remains only to practice entering the number under the root sign!

It's easy!

Let's say we have written down the number

What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!

Why do we need this? Yes, just to expand our capabilities when solving examples:

How do you like this property of roots? Does it make life much easier? For me, that's right! Only we must remember that we can only introduce positive numbers under the square root sign.

Solve this example yourself -
Did you manage? Let's see what you should get:

Well done! You managed to insert the number under the root sign! Let's move on to an equally important one - let's look at how to compare numbers containing the square root!

Comparison of roots

Why should we learn to compare numbers containing the square root?

Very simple. Often, in large and lengthy expressions found on the exam, we get an irrational answer (do you remember what it is? You and I have already talked about this today!)

We need to place the received answers on a coordinate line, for example, to determine which interval is suitable for solving the equation. And here a snag arises: there is no calculator on the exam, and without it how to imagine which number is greater and which is less? That's just it!

For example, define which is greater: or?

You can't tell right off the bat. Well, let's use the analyzed property of entering a number under the root sign?

Then go ahead:

And, obviously, the larger the number under the root sign, the larger the root itself!

Those. if, then,.

From this we firmly conclude that. And no one will convince us otherwise!

Extracting roots from large numbers

Before that, we introduced the factor under the root sign, but how to get it out? You just have to factor it and extract what is extracted!

It was possible to take a different path and decompose into other factors:

Not bad, huh? Any of these approaches is correct, decide what suits you best.

Factoring is very useful when solving non-standard tasks like this:

We are not afraid, but we act! Let us decompose each factor under the root into separate factors:

Now try it yourself (without a calculator! It won't be on the exam):

Is this the end? Don't stop halfway!

That's all, not so scary, right?

Happened? Well done, that's right!

Now try to solve this example:

And an example is a tough nut to crack, so you just can't figure out how to approach it. But we, of course, can tough it.

Well, let's start factoring? Note right away that you can divide a number by (remember the divisibility criteria):

Now, try it yourself (again, without a calculator!):

Well, did it work? Well done, that's right!

Let's summarize

The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal to.
.
If we just take the square root of something, we always get one non-negative result.
Arithmetic root properties:
When comparing square roots, it must be remembered that the larger the number under the root sign, the larger the root itself.

How do you like the square root? All clear?

We tried to explain to you without water everything you need to know on the square root exam.

Now your turn. Write to us whether it is a difficult topic for you or not.

Did you learn something new or everything was already clear.

Write in the comments and good luck on your exams!

In this article, we will cover the main root properties... Let's start with the properties of the arithmetic square root, give their formulations and give proofs. After that, we will deal with the properties of the nth root of the arithmetic.

Page navigation.

Square root properties

At this point, we will deal with the following main properties of the arithmetic square root:

In each of the written equalities, the left and right sides can be swapped, for example, the equality can be rewritten as ... In this "inverse" form, the properties of the arithmetic square root are applied when simplification of expressions as often as in the "direct" form.

The proof of the first two properties is based on the definition of the arithmetic square root and on. And to substantiate the last property of the arithmetic square root will have to be remembered.

So let's start with proof of the property of the arithmetic square root of the product of two non-negative numbers:. For this, according to the definition of the arithmetic square root, it is enough to show that is a non-negative number whose square is equal to a · b. Let's do it. The value of an expression is non-negative as the product of non-negative numbers. The property of the degree of the product of two numbers allows you to write the equality , and since by the definition of the arithmetic square root and, then.

Similarly, it is proved that the arithmetic square root of the product of k non-negative factors a 1, a 2, ..., a k is equal to the product of the arithmetic square roots of these factors. Really, . This equality implies that.

Here are some examples: and.

Now let us prove property of the arithmetic square root of the quotient:. The property of the quotient in natural degree allows us to write the equality , a , and there is a non-negative number. This is the proof.

For example, and .

It's time to take apart property of the arithmetic square root of the square of a number, in the form of equality, it is written as. To prove it, consider two cases: for a≥0 and for a<0 .

Obviously, equality holds for a≥0. It is also easy to see that for a<0 будет верно равенство . Действительно, в этом случае −a>0 and (−a) 2 = a 2. In this way, , as required to prove.

Here are some examples: and .

The property of the square root just proved allows us to substantiate the following result, where a is any real number, and m is any. Indeed, the property of raising a power to a power allows us to replace the power a 2 m by the expression (a m) 2, then .

For example, and .

Properties of the nth root

First, let's list the main properties of n-th roots:

All the written equalities remain valid if the left and right sides are swapped in them. In this form, they are also used often, mainly when simplifying and transforming expressions.

The proof of all the voiced properties of the root is based on the definition of the arithmetic root of the n-th degree, on the properties of the degree and on the definition of the modulus of a number. Let us prove them in order of priority.

Let's start with proof properties of the nth root of the product ... For non-negative a and b, the value of the expression is also non-negative, like the product of non-negative numbers. The property of the product in natural degree allows us to write the equality ... By the definition of an arithmetic root of the nth degree and, therefore, ... This proves the property of the root under consideration.

This property is proved similarly for the product of k factors: for nonnegative numbers a 1, a 2, ..., a n, and .

Here are examples of using the property of the nth root of the product: and .

Let's prove property of the root of the quotient... For a≥0 and b> 0, the condition is satisfied, and .

Let's show examples: and .

Moving on. Let's prove property of the nth root of a number to the nth power... That is, we will prove that for any real a and natural m. For a≥0 we have and, which proves the equality, and the equality obviously. For a<0 имеем и (the last passage is valid due to the property of the degree with an even exponent), which proves the equality, and is true due to the fact that, when talking about the root of an odd degree, we took for any non-negative number c.

Here are examples of using the parsed root property: and .

We pass to the proof of the property of a root from a root. We will swap the places of the right and left sides, that is, we will prove the validity of the equality, which will mean the validity of the original equality. For a non-negative number a, the root of a root of the form is a non-negative number. Remembering the property of raising a degree to a power, and using the definition of a root, we can write down a chain of equalities of the form ... This proves the property of the root from the root under consideration.

The property of a root from a root from a root, etc. is proved in a similar way. Really, .

For instance, and .

Let us prove the following. root exponent shortening property... For this, by virtue of the definition of the root, it is sufficient to show that there is a non-negative number, which, when raised to the power n · m, is equal to a m. Let's do it. It is clear that if the number a is non-negative, then the nth root of the number a is a non-negative number. Wherein , which completes the proof.

Here is an example of using the parsed root property:.

Let us prove the following property - the property of a root of a degree of the form ... Obviously, for a≥0, the degree is a non-negative number. Moreover, its n-th degree is equal to a m, indeed,. This proves the property of the degree under consideration.

For instance, .

Let's move on. Let us prove that for any positive numbers a and b for which condition a , that is, a≥b. And this contradicts the condition a

As an example, we present the correct inequality .

Finally, it remains to prove the last property of the nth root. Let us first prove the first part of this property, that is, we will prove that for m> n and 0 ... Then, due to the properties of a degree with a natural exponent, the inequality , that is, a n ≤a m. And the resulting inequality for m> n and 0
Similarly, by contradiction, it is proved that for m> n and a> 1, the condition is satisfied.

Let us give examples of the application of the proved property of the root in concrete numbers. For example, the inequalities and are true.

Bibliography.

Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for grade 8 educational institutions.

Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginning of analysis: Textbook for 10 - 11 grades of educational institutions.

Gusev V.A., Mordkovich A.G. Mathematics (a guide for applicants to technical schools).