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When solving some mathematical problems, you have to operate with square roots. Therefore, it is important to know the rules of operations with square roots and learn how to transform expressions containing them. The goal is to study the rules of operations with square roots and ways to transform expressions with square roots.

We know that some rational numbers are expressed as infinite periodic decimal fractions, such as the number 1/1998=0.000500500500... But nothing prevents us from imagining a number whose decimal expansion does not reveal any period. Such numbers are called irrational.

The history of irrational numbers dates back to the amazing discovery of the Pythagoreans back in the 6th century. BC e. It all started with a seemingly simple question: what number expresses the length of the diagonal of a square with side 1?

The diagonal divides the square into 2 identical right-angled triangles, in each of which it acts as a hypotenuse. Therefore, as follows from the Pythagorean theorem, the length of the diagonal of a square is equal to

And although Pythagoras and his students did not have a computer, they were the ones who substantiated this fact. The Pythagoreans proved that the diagonal of a square and its side have no common measure (i.e., a segment that would be plotted an integer number of times both on the diagonal and on the side). Therefore, the ratio of their lengths is the number

Following the discovery of the Pythagoreans

How to prove that a number

It remains to conclude that our assumption is incorrect and the rational number m/n is equal to

1. Square root of a number

Knowing the time t , you can find the path in free fall using the formula:

Task . How many seconds will it take for a stone dropped from a height of 122.5 m to fall?

To find the answer, you need to solve the equation

You also have to look for a positive number by its square when solving other problems, for example, when finding the length of a side of a square by its area. Let us introduce the following definition.

Definition . A non-negative number whose square is equal to a non-negative number a is called the square root of a. This number stands for

Thus

Example . Because

You cannot take square roots from negative numbers, since the square of any number is either positive or equal to zero. For example, the expression

zeros: = 10…0

2n zeros n zeros

2n zeros n zeros

For example,

2. Calculating square roots

. Continuing this process, we obtain one after another the digits of the infinite decimal fraction equal to .

and press the key - 8 digits of the value will be displayed on the screen. In some cases it is necessary to use the properties of square roots, which we will indicate below.

If the accuracy provided by a microcalculator is insufficient, you can use the method for refining the value of the root given by the following theorem. Theorem.

If a is a positive number and is an approximate value for by excess, then

Root formulas. Properties of square roots.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."

And for those who “very much…”) In the previous lesson we figured out what a square root is. It's time to figure out which ones exist formulas for roots what are properties of roots

, and what can be done with all this.- this is essentially the same thing. There are surprisingly few formulas for square roots. Which certainly makes me happy! Or rather, you can write a lot of different formulas, but for practical and confident work with roots, only three are enough. Everything else flows from these three. Although many people get confused in the three root formulas, yes...

Let's start with the simplest one. Here she is:

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

What is a square root?

Root formulas. Properties of square roots.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."

This concept is very simple. Natural, I would say. Mathematicians try to find a reaction for every action. There is addition - there is also subtraction. There is multiplication - there is also division. There is squaring... So there is also taking the square root! That's all. This action ( square root) in mathematics is indicated by this icon:

The icon itself is called a beautiful word " radical".

How to extract the root? It's better to look at examples.

What is the square root of 9? What number squared will give us 9? 3 squared gives us 9! Those:

But what is the square root of zero? No problem! What number squared does zero make? Yes, it gives zero! Means:

Got it, what is square root? Then we consider examples:

Answers (in disarray): 6; 1; 4; 9; 5.

Decided? Really, how much easier is that?!

But... What does a person do when he sees some task with roots?

A person begins to feel sad... He does not believe in the simplicity and lightness of his roots. Although he seems to know what is square root...

This is because the person ignored several important points when studying the roots. Then these fads take cruel revenge on tests and exams...

Point one. You need to recognize the roots by sight!

What is the square root of 49? Seven? Right! How did you know it was seven? Squared seven and got 49? Right! Please note that extract the root out of 49 we had to do the reverse operation - square 7! And make sure we don't miss. Or they could have missed...

This is the difficulty root extraction. Square You can use any number without any problems. Multiply a number by itself with a column - that's all. But for root extraction There is no such simple and fail-safe technology. We have to pick up answer and check if it is correct by squaring it.

This complex creative process - choosing an answer - is greatly simplified if you remember squares of popular numbers. Like a multiplication table. If, say, you need to multiply 4 by 6, you don’t add four 6 times, do you? The answer 24 immediately comes up. Although, not everyone gets it, yes...

To work freely and successfully with roots, it is enough to know the squares of numbers from 1 to 20. Moreover there And back. Those. you should be able to easily recite both, say, 11 squared and the square root of 121. To achieve this memorization, there are two ways. The first is to learn the table of squares. This will be a great help in solving examples. The second is to solve more examples. This will greatly help you remember the table of squares.

And no calculators! For testing purposes only. Otherwise, you will slow down mercilessly during the exam...

So, what is square root And How extract roots- I think it’s clear. Now let's find out WHAT we can extract them from.

Point two. Root, I don't know you!

What numbers can you take square roots from? Yes, almost any of them. It's easier to understand what it's from it is forbidden extract them.

Let's try to calculate this root:

To do this, we need to choose a number that squared will give us -4. We select.

What, it doesn't fit? 2 2 gives +4. (-2) 2 gives again +4! That's it... There are no numbers that, when squared, will give us a negative number! Although I know these numbers. But I won’t tell you). Go to college and you will find out for yourself.

The same story will happen with any negative number. Hence the conclusion:

An expression in which there is a negative number under the square root sign - doesn't make sense! This is a forbidden operation. It is as forbidden as dividing by zero. Remember this fact firmly! Or in other words:

You cannot extract square roots from negative numbers!

But of all the others, it’s possible. For example, it is quite possible to calculate

At first glance, this is very difficult. Selecting fractions and squaring them... Don't worry. When we understand the properties of roots, such examples will be reduced to the same table of squares. Life will become easier!

Okay, fractions. But we still come across expressions like:

It's OK. All the same. The square root of two is the number that, when squared, gives us two. Only this number is completely uneven... Here it is:

What's interesting is that this fraction never ends... Such numbers are called irrational. In square roots this is the most common thing. By the way, this is why expressions with roots are called irrational. It is clear that writing such an infinite fraction all the time is inconvenient. Therefore, instead of an infinite fraction, they leave it like this:

If, when solving an example, you end up with something that cannot be extracted, like:

then we leave it like that. This will be the answer.

You need to clearly understand what the icons mean

Of course, if the root of the number is taken smooth, you must do this. The answer to the task is in the form, for example

Quite a complete answer.

And, of course, you need to know the approximate values ​​from memory:

This knowledge greatly helps to assess the situation in complex tasks.

Point three. The most cunning.

The main confusion in working with roots is caused by this point. It is he who gives confidence in his own abilities... Let's deal with this point properly!

First, let's take the square root of four of them again. Have I already bothered you with this root?) Never mind, now it will be interesting!

What number is 4 squared? Well, two, two - I hear dissatisfied answers...

Right. Two. But also minus two will give 4 squared... Meanwhile, the answer

correct and the answer

gross mistake. Like this.

So what's the deal?

Indeed, (-2) 2 = 4. And under the definition of the square root of four minus two quite suitable... This is also the square root of four.

But! In the school mathematics course, it is customary to consider square roots only non-negative numbers! That is, zero and all positive. Even a special term was invented: from the number the sign is called the radical sign (from the Latin “radix” - root), and the number- This non-negative number whose square is the sign is called the radical sign (from the Latin “radix” - root), and the number. Negative results when extracting an arithmetic square root are simply discarded. At school, everything is square roots - arithmetic. Although this is not particularly mentioned.

Okay, that's understandable. It's even better not to bother with negative results... This is not yet confusion.

Confusion begins when solving quadratic equations. For example, you need to solve the following equation.

The equation is simple, we write the answer (as taught):

This answer (absolutely correct, by the way) is just an abbreviated version two answers:

Stop, stop! Just above I wrote that the square root is a number Always non-negative! And here is one of the answers - negative! Disorder. This is the first (but not the last) problem that causes distrust of the roots... Let's solve this problem. Let's write down the answers (purely for understanding!) like this:

The parentheses do not change the essence of the answer. I just separated it with brackets signs from root. Now you can clearly see that the root itself (in brackets) is still a non-negative number! And the signs are result of solving the equation. After all, when solving any equation we must write All Xs that, when substituted into the original equation, will give the correct result. The root of five (positive!) with both a plus and a minus fits into our equation.

Like this. If you just take the square root from anything, you Always you get one non-negative result. For example:

Because it - arithmetic square root.

But if you are solving some quadratic equation, like:

That Always it turns out two answer (with plus and minus):

Because this is the solution to the equation.

Hope, what is square root You've got your points clear. Now it remains to find out what can be done with the roots, what their properties are. And what are the points and pitfalls... sorry, stones!)

All this is in the following lessons.

If you like this site...

By the way, I have a couple more interesting sites for you.)

You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)

You can get acquainted with functions and derivatives.

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 zeros, is equal to the number written by one and 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 do not 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 $y=((x)^(2))$:

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 $y=((x)^(3))$:

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 definitely intersect with our graph. Consequently, the cube root can always be taken 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 degree. 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 most schools 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, 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:

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. With the first expression everything 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.”