Root from an unknown number. Square root. Detailed theory with applications. Square root, arithmetic square root

The area of a square plot of land is 81 dm2. Know yoga side. Let's assume that the length of the side of the square is good X decimeters. Todi area of the house is more expensive X² square decimetres. Shards for the mind, the area is 81 dm², then X² \u003d 81. The length of the side of the square is a positive number. A positive number, the square of which is 81, є is the number 9. When solving problems, it is necessary to know the number x, the square of which is 81, to solve the problem X² \u003d 81. The price has two roots: x 1 = 9 x 2 \u003d - 9, so 9² \u003d 81 і (- 9) ² \u003d 81. Offending numbers 9 і - 9 are called the square roots of the number 81.

Dearly, that one of the square roots X= 9 є positive number. Yogo is called the arithmetic square root of the number 81 and denotes √81, such a rank √81 = 9.

The arithmetic square root of a number A is called a number unknown to me, the square of some old A.

For example, the numbers 6 i - 6 are the square roots of the number 36. When the number 6 is the arithmetic square root of the number 36, the shards 6 are not the number i 62 = 36. The number - 6 is not the arithmetic root.

Arithmetic square root of a number A signified like this: √ A.

The sign is called the sign of the arithmetic square root; A- is called sub-root viraz. Viraz √ A read like this: arithmetic square root of a number A. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In quiet moods, if it is clear that there is an arithmetic root, it will be short: “the square root of A«.

The value of the square root in the warehouse is called the value of the square root. Tsya diya є wrapped up to a square.

It is possible to square the square whether it is a number, but to obtain a square root it is possible not to be a number. For example, it is not possible to draw the square root of the number - 4. Having found such a root, then, having recognized it with a letter X, We would take away the wrong equality x² = - 4, so it’s worth the cost of an unknown number, and on the right - negative.

Viraz √ A maє sens tilki for a ≥ 0. The value of the square root can be written briefly as follows: √ a ≥ 0, (√A)² = A. Equity (√ A)² = A fair for a ≥ 0. In such a way, to change into the fact that the square root of a negative number A dorivnyuє b, then in that √ A =b, it is necessary to reconsider, what are the following two minds: b ≥ 0, b² = A.

Square root of a fraction

Let's count. Respectfully, that √25 = 5, √36 = 6, and it is reversible that equality is victorious.

so yak i , then equanimity is true. Otzhe, .

Theorem: Yakscho A≥ 0 and b> 0, so the root from the fraction is equal to the root from the number book, divided by the root from the banner. It is necessary to bring that: .

Bo √ A≥0 ta √ b> 0, then .

For the yak_styu zvedennya fraction in the foot and the sign of the square root the theorem has been completed. Let's take a look at the sprat of applications.

Calculate, for the finished theorem .

Another butt: Bring what , like A ≤ 0, b < 0. .

Another butt: Calculate.

Reversal of the square root

The guilt of the multiplier z-pіd to the sign of the root. Let Viraz be given. Yakscho A≥ 0 and b≥ 0, then following the root-creation theorem we can write:

Such a transformation is called the guilt of the multiplier of the z-pod sign of the root. Let's look at the butt;

Calculate at X= 2. No middle substitution X= 2 at the root of the viraz to produce a folding calculation. Qi calculation can be forgiven, as if to blame the z-pіd sign of the root multipliers: . Substituting now x = 2 we take:.

Later, with the guilt of the multiplier, the root sign of the sign of the root is a sub-root of the viraz in the vision of creation, in which there is one or more multipliers in the squares of the unknown numbers. Then let's work out the theorem about the root from the extraction and extract the root from the skin multiplier. Let's look at the butt: Forgiveness A \u003d √8 + √18 - 4√2 wines in the first two dodankіv multipliers of the root sign, otrimaєmo:. I encourage you, that jealousy fair only for A≥ 0 and b≥ 0. well A < 0, то .

Let's look at the alignment x 2 = 4. Let's break it down graphically. For cgo, in one coordinate system, we will create a parabola y \u003d x 2 i straight line y \u003d 4 (Fig. 74). The stench is tinted at two points A (- 2; 4) and B (2; 4). The abscissa points A and B are equal to the roots x 2 \u003d 4. Also, x 1 \u003d - 2, x 2 \u003d 2.

Rozmіrkovuyuchi just like that, we know the root equal x 2 \u003d 9 (div. Fig. 74): x 1 \u003d - 3, x 2 \u003d 3.

And now let's try rozv'yazati equal x 2 \u003d 5; geometric illustrations are presented in fig. 75. It is clear that there are two roots x 1 and x 2, moreover, q numbers, like i in two forward slopes, are equal for the absolute value and prolongation for the sign (x 1 - x 2) - Ale on the front of the front slopes, de root equals were easily found (because they can be known without peeling graphs), with equals x 2 = 5 on the right is not so: we cannot show the meaning of the roots behind the armchairs, we can only establish that one root is rooted in three lions more points - 2 , and the other is three times right

Points 2.

What is the number (point), how do the three right-handed points 2 and how squared give 5? Zrozumіlo, sho tse 3, oskіlki Z 2 = 9, i.e. go out more, lower it is necessary (9\u003e 5).

So, for us, the number is spread between the numbers 2 and 3. But between the numbers 2 and 3, there are impersonal rational numbers, for example and so on. It is possible that there is such a friend among them, what? We won’t have the same problems from equals x 2 - 5, we can write what

Ale, here we are in for an unacceptable surprise. It appears, there is no such fraction, for which jealousy wins
The proof of the formulated assertion is foldable. Tim is not smaller, we are guided by yoga, the shards are more beautiful and at the back, even better to try yoga intellect.

It is acceptable that such a short-lived drіb, on the yak vykonuєtsya equanimity. Then, then m2 = 5n2. Remaining equality means that the natural number m 2 is divisible without excess by 5 (for private view n2).

Later, the number m 2 ends with the number 5, the number 0. But the natural number m ends with the number 5, the number 0, then. the number m is divisible by 5 without excess. Otherwise, it seems that if the number m is subdivided by 5, then the private viide is a natural number k. Tse means
that m = 5k.
And now wonder:
m 2 \u003d 5n 2;
Imagine 5k zam_st m for pershu equanimity:

(5k) 2 = 5n 2, then 25k 2 = 5n 2 or n 2 = 5k 2 .
Remaining jealousy means that the number. 5n 2 is divisible by 5 without excess. Rozmіrkovuchi, like even more, we come to the visnovka about those that the number n is divisible by 5 without excess.
Also, m is divided by 5, n is divided by 5, later, drіb can be short (by 5). And then we allowed that the drib was not short. Why is it on the right? Why, rightly mirkuyuchi, we came to the point of absurdity, or, as mathematicians often say, took away the wipe "!
Zvіdsi robimo visnovok: there is no such fraction.
The method of proof, which we have stubbornly stumbled upon, is called in mathematics the method of proving the protivolego. The essence of yoga offensive. It is necessary for us to bring firmness to the deacon, but we allow it to be unacceptable (mathematicians seem: “tolerably unacceptable” - not in sensi “unacceptable”, but in sensi “as far as it is necessary”).
If, as a result of legal mirkuvan, we come to super-accuracy with the mind, then we are robbed of whiskers: our admission is wrong, then, those who needed to be brought to it were correct.

Later, only rational numbers are possible (and we don’t know other numbers yet), equal x 2 \u003d 5 is not possible to overcome.
Having studied ahead with a similar situation, mathematicians realized that it was necessary to come up with a way to describe my mathematical language. They introduced a new symbol to the point of view, which they called the square root, and for the additional symbol of the root equal x 2 \u003d 5 they wrote it down like this:

it is expected: "the square root of z 5"). Now, for any kind of equal mind, x 2 \u003d a, de a\u003e O, you can know the root - they are numbers , (Mal. 76).

More heavenly support, scho the number is not whole and not even.
It means that it is not a rational number, but the number of a new nature, about such numbers we will specially talk later, divided 5.
For the time being, it is less significant, but the new number is between the numbers 2 and 3, the shards 2 2 = 4, and less, lower 5; Z 2 \u003d 9, and ce more lower 5. You can specify:

True, 2.2 2 = 4.84< 5, а 2,3 2 = 5,29 >5. You can
specify:

really, 2.23 2 = 4.9729< 5, а 2,24 2 = 5,0176 > 5.
In practice, it’s important to note that the number is more expensive 2.23, or it’s more expensive 2.24, but it’s not just jealousy, but jealousy is close, for the recognition of such a victorious symbol.
Otzhe,

Discussing the solution of equal x 2 \u003d a; Poppying in Non -standard, nezhtetnu (yak to love cosmonauty) the situation I did not know the vichens of the non -ethnicity of the udomikh, the mathematics for mathematical models, and the nobility of the Termin ibe nobility (new symbol); in other words, stink to introduce a new understanding, and then increase the power of that
concepts. Tim himself, the new understanding of this yoga understanding is becoming the head of the Mathematical Movement. We did it the same way: they introduced the term “square root of the number a”, introduced a symbol for its meaning, and three years to win the power of a new concept. So far, we only know one thing: that a > 0,
then - a positive number that satisfies the equality x 2 \u003d a. In other words, this is a positive number, when squared, the number a comes out.
Oskilki equal x 2 \u003d 0 maє root x \u003d 0
Now we are ready to give a reading of the appointment.
Appointment. The square root of an unknown number is called such an unknown number, the square of some old number.

Tse number is meant, and the number at which is called the root number.
Otzhe, as if a is not a number, then:

Yakscho a< О, то уравнение х 2 = а не имеет корней, говорить в этом случае о квадратном корне из числа а не имеет смысла.
In this rank, viraz maє sense less for a > 0.
Say what - one and the same mathematical model (one and the same staleness between unknown numbers
(and that b), but only a friend is described by more simple mine, lower first (vicory simple symbols).

The operation of finding the square root of a negative number is called the change of the square root. Tsya operation is a reversal by bringing to life in the square. Level:

Once again, respect: the tables have less positive numbers, the shards are not assigned to the designated square root. I want, for example, (- 5) 2 \u003d 25 - the equality is correct, go to the next entry with the square root of the variant (so write what.)
can't. For the apology, . - A positive number means .
Often say not "square root", but "arithmetic square root". The term "arithmetic" is omitted for the sake of style.

D) On the view of the front butts, we can indicate the exact value of the number. It was less clear that it was bigger, lower 4, ale smaller, lower 5, oskolki

42 = 16 (smaller, lower 17), and 52 = 25 (higher, lower 17).
Vtіm, the nearest value of the number can be known for the help of a microcalculator, how to avenge the operation of the square root; the value is more expensive 4.123.
Otzhe,
The number, like and look at the number is not rational.
e) It is not possible to calculate, the square root of a negative number cannot be used; a record of indulgences to the sens. The order was proponated incorrectly.
e) , oskіlki 31 > 0 і 31 2 = 961. In such cases, you can win the table of squares of natural numbers and a microcalculator.
g), shards 75 > 0 and 75 2 = 5625.
In the simplest cases, the values of the square root are counted in a row: meager. bud. In folding situations, it is necessary to bring up a table of squares of numbers chi and carry out calculations with an additional microcalculator. And how buti, how can one hand no tables, no calculator? V_dpovіmo on the chain of food, virіshivshi attacking the butt.

butt 2. Calculate
Solution.
First stage. It doesn't matter if you guess that the vidpovid viide has 50 іz "tail". In fact, 50 2 = 2500, and 60 2 = 3600, and the number 2809 is changed between the numbers 2500 and 3600.

Another stage. We know the "tail", tobto. I will leave the figure of the stupid number. As long as we know that the root is growing, then in the future you can have 51, 52, 53, 54, 55, 56, 57, 58 or 59. You only need to check two numbers: 53 and 57, the smell of stench when squared will give b in The result is a different number that ends with the number 9, then the same number that ends with the number 2809.
Maєmo 532 = 2809 tse those that we need (we were lucky, we were squandered in the "apple"). Otzhe, = 53.
Suggestion:

53
example 3. The legs of a straight-cut tricutnik are 1 cm and 2 cm thick. Why is the tricutnik hypotenuse? (Mal.77)

Solution.

We quickly follow the geometry of the Pythagorean theorem: the sum of the squares of the lengths of the legs of a straight-cut tricot is equal to the square of the length of its hypotenuse, so a 2 + b 2 \u003d c 2 de a, b - legs, c - hypotenuse of a straight-cut tricot.

To mean,

This butt shows that the introduction of the square root is not a mathematician's bug, but an objective necessity: in real life, there are situations, mathematical models of which can overcome the operation of forcing the square root. Maybe, the most important of such situations is related to
rozvyazannyam square rivnyan. Dosi, using square equals ax 2 + bx + c \u003d 0, we either laid out the left part into multipliers (which turned out to be far from a reality), or they scored graphic methods (which are not too fancy, but beautiful). Truly for visualization
root x 1 and x 2 of the square equation ax 2 + bx + c = 0

revenge, as you can see, the sign of the square root. Qi formulas zastosovuyutsya practically in such a rank. Come on, for example, you need to split 2x 2 + bx - 7 = 0. Here a = 2, b = 5, c = - 7. Later,
b2 - 4ac \u003d 5 2 - 4. 2. (- 7) \u003d 81. Dali is known. To mean,

More we have designated, which is not a rational number.
Mathematicians call such numbers irrational. Irrational - be it a number mind, as if the square root does not appear. For example, and etc. - Irrational numbers. In 5 reports, we will talk about rational and irrational numbers. Rational and irrational numbers at once become impersonal real numbers, that is. impersonal numbers, with which we can operate in real life (for
news). For example, all these are valid numbers.
Likewise, as we have already designated the concept of the square root, we can assign the concept of the cube root: the cube root of an unknown number a is called a number that is unknown to me, the cube of which is a number. Otherwise, apparently, jealousy means that b 3 \u003d a.

Everything is possible in the course of algebra of the 11th grade.

The concept of the square root of an unknown number

Let's look at the alignment x2 = 4. Let's break it down graphically. For whom in one system coordinates zbuduєmo parabola y \u003d x2 i straight line y \u003d 4 (Fig. 74). The stench is tinted at two points A (- 2; 4) and B (2; 4). The abscissa points A and B are equal to the roots x2 = 4. Also, x1 = - 2, x2 = 2.

Razmirkovuyuchi so it is, we know the root equal x2 = 9 (div. fig. 74): x1 = - 3, x2 = 3.

And now let's try rozv'yazati equal x2 = 5; geometric illustrations are presented in fig. 75. It is clear that there are two roots x1 and x2, moreover, the number of numbers, like and in two forward slopes, is equal for the absolute value and the length behind the sign (x1 - - x2) babysitter if you could easily find them (because you could know them without being scraggly with graphs), if x2 = 5 on the right, it’s not like that: we can’t show the meaning of the roots behind the armchairs, we can only put it in one root three points to the left of the point - 2, and the other - three to the right of the point 2.

Ale, here we are in for an unacceptable surprise. Appear, there is no such fractions DIV_ADBLOCK32">

It is acceptable that it is such a short-lived drіb, for which equanimity is victorious https://pandia.ru/text/78/258/images/image007_16.jpg" alt=".jpg" width="55" height="36">!}!}, i.e. m2 = 5n2. Remaining jealousy means that natural number m2 can be divided without excess by 5 (private wide has n2).

Later, the number m2 ends with the number 5, the number 0. But the natural number m ends with the number 5, the number 0, i.e. the number m is divided by 5 without excess. Otherwise, it seems that if the number m is subdivided by 5, then the private viide is a natural number k. Ze means that m = 5k.

And now wonder:

Imagine 5k zam_st m for pershu equanimity:

(5k) 2 = 5n2, then 25k2 = 5n2 or n2 = 5k2.

Remaining jealousy means that the number. 5n2 is divided by 5 without excess. Rozmirkovuchi, like more, we come to the visnovka about those that the number n is divisible by 5 without surplus.

Also, m is divided by 5, n is divided by 5, later, drіb can be short (by 5). And then we allowed that the drib was not short. Why is it on the right? Why, rightly mirkuyuchi, we came to the point of absurdity, or, as mathematicians often say, took away the wipe "! ).

If, as a result of legal mirkuvan, we come to super-accuracy with the mind, then we are robbed of whiskers: our admission is wrong, then, those who needed to be brought to it were correct.

Father, floating in your order only rational numbers(And we still don’t know the other numbers), equal x2 = 5 and we can’t beat it.

Having studied ahead with a similar situation, mathematicians realized that it was necessary to come up with a way to describe my mathematical language. They introduced a seemingly new symbol, which they called the square root, and for the additional symbol of the root equal x2 \u003d 5 they wrote it down like this: ). Now, for whatever reason, x2 = a, de a > Oh, you can know the root - they are numbers https://pandia.ru/text/78/258/images/image012_6.jpg" alt=".jpg" width="32" height="31">!}!} not healthy and not dry.
It means that it is not a rational number, but the number of a new nature, about such numbers we will specially talk later, divided 5.
For the time being, it is less significant, but the new number is between the numbers 2 and 3, the shards 22 = 4, and less, lower 5; Z2 \u003d 9, and more lower than 5. You can specify:

Once again, respect: the tables have less positive numbers, the shards are not assigned to the designated square root. If, for example, = 25 - equalness is correct, go to the next entry to the record of the square root (to write what). .jpg" alt=".jpg" width="42" height="30">!}!}- A positive number means https://pandia.ru/text/78/258/images/image025_3.jpg" alt=".jpg" width="35" height="28">!}!}. It was more reasonable that it was bigger, lower 4, ale, smaller, lower 5, 42 = 16 (smaller, lower 17), and 52 = 25 (less larger, lower 17).
Vtіm, the nearest value of the number can be known for help microcalculator how to avenge the square root operation; the value is more expensive 4.123.

The number, like and look at the number is not rational.
e) It is not possible to calculate, the square root of a negative number cannot be used; a record of indulgences to the sens. The order was proponated incorrectly.
e) https://pandia.ru/text/78/258/images/image029_1.jpg" alt="Zavdannya" width="80" height="33 id=">!}!} shards 75 > 0 і 752 = 5625.

In the simplest cases, the values of the square root are counted in multiples:

https://pandia.ru/text/78/258/images/image031_2.jpg" alt="Zavdannya" width="65" height="42 id=">!}!}
Solution.
First stage. It doesn't matter if you guess that the vidpovid viide has 50 іz "tail". In fact, 502 = 2500, and 602 = 3600, and the number 2809 is changed between the numbers 2500 and 3600.

Glancing once more at the sign... And let's go!

Let's start from a simple one:

Khvilinka. tse, and tse means that we can write it like this:

Conquered? The axis of your advance:

The root of the numbers, what happened, don't really stand out? Do not bіda - the axis of you so apply:

And how many multipliers are not two, but more? Same! The formula for the multiplication of roots works with whether there are any number of multipliers:

Now I will do it myself:

Suggestions: Well done! Wait, everything is easy, you know the multiplication table!

Rozpodіl koreniv

We have taken many roots, now let's get down to power.

I’ll guess that the formula for the infamous looks like this:

What does it mean root from a part of a private root.

Well, let's take a look at the butts:

Axis i all science. And the axis is such an example:

Everything is not so smooth, like a first butt, ale, like a bachish, there is nothing folding.

And what, how to get drunk such a viraz:

It is necessary to simply zastosuvat formula at the gate directly:

And the axis is such an example:

Can you see such a viraz:

All the same, only here you need to guess, how to shift the fractions (if you don’t remember, look at the topic and turn around!). Guessing? Now we see it!

Enraptured, that you are with us, we have run into, now we will try to root the world.

Zvedennya in the foot

And what will you do, like a square root to square? It's simple, we guess the sense of the square root of a number - the whole number, the square root of some kind.

So from, how do we create a number, the square root of a certain number, a square, then what is taken?

Well, it's awesome!

Let's take a look at the examples:

Everything is simple, right? And what will be the root of another world? Nothing terrible!

Seek out those logics and remember the power and the ability to step by step.

Read the theory on the topic "" and you will become extremely clear.

Axis, for example, such a viraz:

Whose butt will have male feet, but what will the wine be unpaired? Well, I know, stop the level of power and spread everything into multipliers:

From this point everything is clear, but how to win the root of the number in the world? Axis, for example:

Easy to drink, right? And what about more than two steps? Dorimuёmosya ієї zh logic, vikoristuyuyuchi power steps:

Well, how did everyone understand? Apply these same verses yourself:

A axis i vіdpovіdі:

Introduced pid root sign

Why haven’t we learned how to work with roots! It only took a little time to try to enter the number of roots!

It's too easy!

Suppose we have a number

What can we do with him? Well, zvichayno, close the trinity under the root, remembering at the same time that the triplet is the square root!

What else do we need? It's so simple, to expand our possibilities with perfect applications:

How is that power of the root? Is it really a question of life? On me, that's right! Tilki Keep in mind that we can only add a square root sign to a positive number.

Virish independently the axis of the butt -
Rushed? Let's marvel, what can you see in you:

Well done! You have far enough to enter the number pіd sign of the root! Let's move on to something that is not less important - let's look at how to correct the numbers to revenge the square root!

Root repair

How about we learn to figure out the numbers, how to avenge the square root?

Kind of simple. Often, at the great and trivial virazas, who speak in sleep, we take irrational evidence (remember, what is it like that? We were already talking about you today!)

Otrimani vіdpovіdі we need to spread out on the coordinate line, for example, to determine which interval is suitable for rozvyazuvannya rivnyannya. The first axis here blames the zakovik: there is no calculator in use, but without it, how to reveal, which number is larger, and which is smaller? Otozh i out!

For example, vyznach, what is more: chi?

You won’t tell right away. Well, what, is it quick to draw the power of the introduced number under the sign of the root?

Go ahead:

Well, obviously, the larger the number under the sign of the root, the larger the root itself!

Tobto. yakscho, otzhe, .

Zv_dsi firmly robimo visnovok, scho. And no one can change us from the other side!

Foreshadowing the Roots of Great Numbers

Before whom did we introduce the multiplier under the sign of the root, but how can I blame it? You just need to lay out yogo on multipliers and pull those who are pulling!

It was possible to drink with a different way and spread it on other multipliers:

Not bad, right? Be-yaky іz tsikh podkhodіv vіrniy, virіshuy like you handily.

Arrangement for multipliers will be in good fortune with the implementation of such non-standard tasks, like the axis of the chain:

Do not lakaєmos, but diemo! We put together a leather multiplier under the root on an okremi multiplier:

And now try it on your own (without a calculator! You won’t be able to sleep on yoga):

Hiba tse kinets? Don't be fooled by pivdoroz!

Axis and everything, not so everything and scary, right?

Wiishlo? Well done, you're right!

And now try this butt of virishiti:

And the butt is a mitzny pot, so you won’t be able to pick it up right away, like you’ll step up to a new one. Ale us wines, obviously, in the teeth.

Well, how about arranging for multipliers? It is highly respectful that you can add the number to (we guess the signs of divisibility):

And now, try it yourself (I know, without a calculator!):

Well scho, wiyshlo? Well done, you're right!

P_vedemo p_bags

The square root (arithmetic square root) of an unknown number is called such an unknown number, the square of some other number.
.
If we simply take the square root of everything, then we always take one invisible result.
Power of the arithmetic root:
When the square root is equal, it is necessary to remember that the greater the number under the sign of the root, the greater the root itself.

How is your square root? Has everything made sense?

We have tried to explain to you without driving everything that is necessary to know in sleep about the square root.

Now your devil. Write to us a suitable topic for you.

Recognizing you now, everything was so clear.

Write in the comments and good luck on your sleep!

At tsіy statti mi zaprovadimo understand the root of the number. Dyatimemo sequentially: starting from the square root, let's move on to the description of the cubic root, after which we can understand the root, denoting the root of the n-th degree. At the same time, it introduces a name, a sign, suggests an application of roots and gives the necessary explanations for that comment.

Square root, arithmetic square root

To understand the meaning of the root of the number, and the square root of the zokrem is necessary for the mother. At this point, mi often zishtovhuvatimosya with another step of the number - the square of the number.

Pochnemo s square root denominator.

Appointment

Square root of a- Tse number, the square of some old a.

Schob lead apply the square root, Let's take some numbers, for example, 5 , −0.3 , 0.3 , 0 (−0.3) 2 =(−0.3) (−0.3)=0.09, (0.3) 2 = 0.3 0.3 = 0.09 i 0 2 = 0 0 = 0). Then, for given assignments, the number 5 is the square root of the number 25, the numbers −0.3 and 0.3 are the square roots of 0.09, and 0 is the square root of zero.

Slid designate, for whatever number a іsnuє, the square of koho dorivnuє a. And for itself, for any negative number a, do not use the same decimal number b, the square of any other number a. True, equality a=b 2 is impossible for any negative a , shards b 2 - I don’t know the number for any b . in such a manner, on impersonal real numbers there is no square root of a negative number. In other words, on impersonal real numbers, the square root of a negative number does not stand out and does not make sense.

Sounds like a logical food: “And what is the square root of a for whether there is a lot of a”? Vidpovid - so. Grounded on this fact, a constructive method is important in order to win the significance of the value of the square root.

Then put forward a more logical reason: “What is the number of all square roots of a given infinite number a - one, two, three, more more”? Axis v_dpov_d on new: if a is equal to zero, then the single square root of zero is zero; for example, a is a positive number, the number of square roots from the number a is equal to two, moreover, the root is є. Obguruntuemo tse.

Goodbye a=0 . On the other hand, it is shown that zero is true by the square root of zero. The reason for the obvious evenness 0 2 =0 0=0 is the designation of the square root.

Now we can say that 0 is the single square root of zero. Speeding up by the method of seeing the unacceptable. Let's assume that the number b is known to be the same number as zero, but it is the square root of zero. Todi maє vykonuvatisya umova b 2 =0, which is impossible, shards for be-yakom vіdminnym vіd zero b value virazu b 2 є positive. We didshli super-sharpness. It is necessary to bring that 0 is the single square root of zero.

We pass to vipadkіv, if a is a positive number. We were told more, that you have to use the square root of any number, let the square root a equal the number b. It is acceptable that є is the number c, but also є is the square root of a. Then the square root of the fairness b 2 \u003d a і c 2 \u003d a, їх sli, sho b 2 − c 2 \u003d a−a \u003d 0, but the shards b 2 − c 2 \u003d (b−c) ( b + c ) , then (b-c) · (b + c) = 0 . Jealousy is taken away from strength powers dіy іz dіysnimi numbers perhaps only then, if b-c=0 or b+c=0. In this order, the numbers b and c are equal or protilege.

If we allow that the number d, with one more square root at the warehouse a, then by the mirroring, similar to the ones we have already pointed, it should be brought, that d is closer to the number b or to the number c. Also, the number of square roots from a positive number is equal to two, moreover, the square root is opposite numbers.

For efficiency of work with square roots, the negative root is “reinforced” as a positive one. Z tієyu method to be introduced derivation of the arithmetic square root.

Appointment

The arithmetic square root of a negative number a- Tse nevіd'єmne number, the square of which dovnyuє a.

For the arithmetic square root of warehouse a, the value is taken. The sign is called the arithmetic square root sign. Yogo is also called the sign of the radical. This can be partly a little like a “root”, and also a “radical”, which means the same object.

The number under the sign of the arithmetic square root is called root number, and viraz under the sign of the root - subroot virazom, in their term "sub-root number" is often replaced by "sub-root number viraz". For example, in the entry, the number 151 is the main root number, and in the entry viraz a, the root is viraz.

When reading, the word "arithmetic" is often omitted, for example, the record is read as "square root of seven twenty nine cent". The word "arithmetic" is used only once, if you want to be especially blatant, you can go about the positively square root of the number.

At the light of the introduced value, the arithmetic square root of the arithmetic square root has the same value as any non-negative number a.

The square root of a positive number a behind the additional sign of the arithmetic square root is written as i. For example, the square root of the number 13 є i. The arithmetic square root of zero is equal to zero, then . For negative numbers a, the entries mi are not subject to sensation until the event complex numbers. For example, to relieve the sense of expression that.

For subbags of the significance of the square root, the power of the square roots is brought to the fore, which is most likely to be practical.

At the end of this point, it is worth respecting that the square root of the number a є solutions to the form x 2 \u003d a better change x.

Cubic root of number

Definition of the cube root warehouse a is given in the same way as the square root. It is only based on the understanding of the cube of the number, but not the square.

Appointment

The cube root of the number a the number is called, the cube of which is equal to a.

Navigable apply a cubic root. For which number of numbers, for example, 7 , 0 , −2/3 i know їх y cube: 7 3 =7 7 7=343 , 0 3 =0 0 0=0 , . So, basing on the designation of the cube root, you can confirm 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.

You can show that the cube root of the warehouse a, on the square root, zavzhdi іsnuє, moreover, for non-negative a , but for any real number a. For whom you can win the very same way, about which we guessed the square root.

Moreover, there is no longer a single cube root for a given number a. We bring the rest of the firmness. In this context, we can see three vipadas: a is a positive number, a=0 and a is a negative number.

It is easy to show that if the cube root of a is positive, it cannot be either a negative number or zero. True, let b є a cubic root for a, then for the same we can write equality b 3 \u003d a. Apparently, the certitude can be correct for negative b і for b=0 , the spikes for the negatives b 3 =b·b will be a negative number chi zero obviously. Also, the cubic root of a positive number a is a positive number.

Now it is acceptable that the number b has one more cubic root from the number a, significantly one c. Then c 3 = a. Later, b 3 −c 3 =a−a=0 , but b 3 −c 3 =(b−c) (b 2 +b c+c 2)(the formula for short multiplication difference of cubes), stars (b−c) (b 2 +b c+c 2)=0 . Otriman's jealousy is only possible if b−c=0 or b 2 +b c+c 2 =0 . From the first equality, b=c is possible, and there is no other solution, because the left part is a positive number for any positive numbers b і c as the sum of three positive additions b 2 , b c і c 2 . Cim brought the unity of the cube root of a positive number a.

When a=0, the cube root of warehouse a є is more than the number zero. It is clear that if you assume that the number b is used, if you see zero as a cube root from zero, then the equality of b 3 \u003d 0 is to blame, as it is only possible with b \u003d 0.

For negative a, you can induce a mirroring, similar to the positive a. First, it is shown that the cube root of a negative number cannot equal a positive number, nor zero. In a different way, let's assume that there is another cubic root from a negative number and it is shown that the wines of the language are combined with the first.

Otzzhe, zavzhd іsnuіє korіnіch s of any given decimal number a, moreover, one.

Damo designation of the arithmetic cube root.

Appointment

The arithmetic cube root of an infinite number a a number is called unknown to me, a cube of some old a.

The arithmetic cube root of the unknown number a is indicated as a sign called the sign of the arithmetic cube root, the number 3 in this record is called root indicator. The number under the sign of the root - tse root number, viraz under the sign of the root - tse subroot viraz.

If you want the arithmetic cube root to be assigned only negative numbers a, you can also manually win the entries, for which the sign of the arithmetic cube root changes the negative numbers. Let's sum it up like this: , de a is a positive number. For example, .

We will talk about the power of the cubic root in the main article of the power of the roots.

The calculation of the value of the cube root is called the calculation of the cube root, the reason is taken from the article of the hero of the roots: ways, apply, solutions.

At the end of this paragraph, let's say that the cube root of the warehouse is a є solutions to the form x 3 =a.

Root of the nth stage, arithmetic root of the stage n

It’s easy to understand the root of the number - we introduce designation of the root of the n-th stage for n.

Appointment

The root of the nth degree of the number a- Tse number, n-th step of what is more expensive a.

From which appointment it was understood that the root of the first stage of the number a is the number a, the shards of the same stage with the natural indicator were taken a 1 \u003d a.

We have looked more closely at the n-th degree root slopes at n=2 and n=3 – the square root and the cube root. So the square root is the root of another level, and the cube root is the root of the third level. To extract the roots of the n-th step with n=4, 5, 6, ... їх manually divide them into two groups: the first group is the root of the paired steps (tobto, with n=4, 6, 8, ...), the other group is the root of the unpaired steps (tobto, at n=5, 7, 9, …). Therefore, the root of the paired steps is similar to the square root, and the root of the unpaired steps is cubic. Let's sort them out with them.

Let's look at the roots, the steps of which are the guys of the number 4, 6, 8, ... As we have already said, the stench is similar to the square root of the number a. That is the root of any paired step from the number a іsnuє only for not much a. Moreover, if a=0, then the root a is single and equal to zero, and if a>0, then there are two roots of the paired step from the number a, moreover, they are opposite numbers.

Obguruntuemo remains hardened. Let b be the root of the paired degree (significantly її yak 2m, de m is a natural number) from the number a. Assume that the number c is one more root of the step 2·m in warehouse a. Then b 2 m −c 2 m =a−a=0 . We know the form b 2 m − c 2 m = (b − c) (b + c) (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2) then (b−c) (b+c) (b 2 m−2 +b 2 m−4 c 2 +b 2 m−6 c 4 +…+c 2 m−2)=0. Z ієї іїї іїї vіplivaєє, scho 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 equals mean that the numbers b and c are equal or b and c are protileges. And the rest of the equality is fair only for b = c = 0, the shards of the left part of the left part are virazed, as it is non-negative for any b and as the sum of non-negative numbers.

As for the roots of the n-th degree with unpaired n, then the stench is similar to the cubic root. So the root of any unpaired degree from the number a is used for any decimal number a, moreover, for a given number a vіn є єdine.

The unity of the root of the unpaired step 2 m+1 in warehouse a is brought by analogy with the proof of the unity of the cube root of a. Only here is the deputy of jealousy a 3 −b 3 =(a−b) (a 2 +a b+c 2) victoriousness 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). Viraz in the rest of the arc can be rewritten like 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, at m=2 maybe 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)). If a and b are offensive positives and negative negatives are a positive number, then viraz b 2 +c 2 +b·c, which is in the arms of the highest level of investment, is positive as the sum of positive numbers. Now, protruding sequentially up to the viraz at the arches of the forward steps of investment, we switch over, that the stench is also positive as a sum of positive numbers. It is necessary for the result 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 It is possible only once, if b−c=0, then if the number b is equal to the number c.

The time has come to explore the roots of the n-th level. For whom is it given designation of the arithmetic root of the nth degree.

Appointment

The arithmetic root of the nth degree of an infinite number a the number is called unknown to me, the n-th step of some kind of a.