The concept of the square root of negative quantity
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). Abscissa point A i є roots equal 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.
Now we will try virishity 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 for the sign (x1 - - x2) if they were known without practice (because they could be known and not covered with graphs), with x2 = 5 on the right, it’s not like that: behind the chairs we can’t show the meaning of the roots, we can only put it in, just 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="(!LANG:(!LANG:.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 naturally the 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:
Assume 5k instead of 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.
Otzhe, m is subdivided by 5, n is subdivided by 5, also, drib can be short (by 5). And then we allowed that the drib was not short. Why is it on the right? Why, correctly mirkuyuchi, we came to the point of absurdity, or, as mathematicians often seem, they took off the rubbish "! ).
If, as a result of the correct mirkuvan, we come to superbility with the mind, then robimo the whiskers: our pardon is unverifiable, then, we believe those that it was necessary to bring.
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 guess how to describe mathematical language. They introduced a new symbol to the point of view, which they called the square root, and with the aid of this root symbol 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 numbershttps://pandia.ru/text/78/258/images/image012_6.jpg" alt="(!LANG:(!LANG:.jpg" width="32" height="31">!}!} not healthy and not dry.
Later, not a rational number, but the number of a new nature, we will specially talk about such numbers later, at the division 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 only positive numbers; If, for example, = 25 - equalness is correct, go to the next entry to the record of the square root (to write what). .jpg" alt="(!LANG:(!LANG:.jpg" width="42" height="30">!}!}- a positive number, https://pandia.ru/text/78/258/images/image025_3.jpg" alt="(!LANG:(!LANG:.jpg" width="35" height="28">!}!}. It was more reasonable that it was more, lower 4, and less, lower 5, since 42 = 16 (lower, lower 17), and 52 = 25 (higher, 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 task was proponated incorrectly.
e) https://pandia.ru/text/78/258/images/image029_1.jpg" alt="(!LANG:(!LANG:Zavdannya" width="80" height="33 id=">!}!}, Oskilki 75 > 0 and 752 = 5625.
In the simplest cases, the values of the square root are calculated once:
https://pandia.ru/text/78/258/images/image031_2.jpg" alt="(!LANG:(!LANG: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 between the numbers 2500 and 3600.
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 that x 2 \u003d - 9, shards 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 is √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, but on the right it’s 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 revise that such two think: 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 shot at the feet 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 fault of the multiplier 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:.
Otzhe, with the fault of the multiplier z-pіd of the sign of the root є sub-root viraz at the creation, in which there is one or more multipliers in є squares of unknown numbers. Then let's work out the theorem about the roots of creation and draw the roots from the skin multiplier. Let's look at the butt: Forgiveness A \u003d √8 + √18 - 4√2 in the first two dodankiv multipliers of the root sign, we take:. 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). Abscissa point A i є roots equal 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.
Now let's try virishity equal x 2 = 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 length for the sign (x 1 - x 2) De root equals were known without practice (because they can be known and not hardened by graphs), with equals x 2 \u003d 5 on the right is not so: we cannot show the meaning of the roots behind the armchairs, we can only set that one root is rooted three points - 2, and the other one is a little more 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. Perhaps 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 to be completed. 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;
Assume 5k instead of 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.
Otzhe, m is subdivided by 5, n is subdivided by 5, also, drib can be short (by 5). And then we allowed that the drib was not short. Why is it on the right? Why, correctly mirkuyuchi, we came to the point of absurdity, or, as mathematicians often seem, they took off the rubbish "!
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 is coming. 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 the correct mirkuvan, we come to superbility with the mind, then robimo the whiskers: our pardon is unverifiable, then, we believe those that it was necessary to bring.
Otzhe, looming over rational numbers (and we don’t know other numbers yet), equal x 2 \u003d 5 is not possible for us.
Having studied ahead with a similar situation, mathematicians realized that it was necessary to guess how to describe mathematical language. They introduced a seemingly new symbol, 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.
Later, not a rational number, but the number of a new nature, we will specially talk about such numbers later, at the division 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 one is 2.23, or else it’s more expensive 2.24, but it’s not just equal, but equal is close, for the recognition of what a victorious symbol is.
Otzhe, 
Discussing the solution of equal x 2 \u003d a; Spend time in a non-standard, out-of-the-ordinary situation (like to love the cosmonauts who are fluttering) and not knowing how to get out of it for additional help, mathematicians predict for a mathematical model, which was previously used, a new term and a new meaning (new symbol); otherwise, apparently, they introduce a new understanding, that buv will increase the power of that
concepts. Tim himself, the new understanding of this yoga understanding is becoming the head of the Mathematical Movement. We acted the same way: they introduced the term “square root of the number a”, introduced a symbol for its meaning, and three years for 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 a non-negative number a is a non-negative number, the square of which is equal to a.
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 is less than > 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 only positive numbers; 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 it seems 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’s less clear that it’s bigger, lower 4, ale less, lower 5, oscalki
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 task 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 calculated twice: and so on. And how buti, how can one hand no tables, no calculator? Vidpovіmo tse pitannya, virishivshi offensive 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 \u003d 2500, and 60 2 \u003d 3600, the number 2809 is 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. Only two numbers need to be checked: 53 and 57 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 once consumed to 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? (Fig.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 that 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 problem, but an objective necessity: in real life, situations are becoming more common, mathematical models of which can overcome the operation of forcing the square root. Maybe, the most important of such situations is related to
rozvyazuvannyam square rivnyan. Dosi, using square equals ax 2 + bx + c = 0, we either laid out the left part into multipliers (which turned out to be far from a reality), or victorious graphic methods (which are not too fancy, but beautiful). Really for a joke
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 = 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 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. In other words, equanimity means that b3 = a.
Everything is possible in the course of algebra of the 11th grade.
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, i of the square root, zokrema, mother needs. 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.
Next, designate that for whatever number a is the square of some kind of 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. Based on this fact, one can introduce a constructive method, which is used to determine the value of the square root.
Do you post a more logical reason: “What is the number of all square roots of a given infinite number a - one, two, three, or even more”? Axis vіdpovіd on new: if a is equal to zero, then the single square root of zero is zero; if a is a positive number, then 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 equality is 02 = 00 = 0 and the square root is chosen.
Now we can say that 0 is the single square root of zero. Speeding up by the method of seeing the unacceptable. Let's say that it is the number b, which is equal to zero, and that it is the square root of zero. Then it is possible to win the mind b 2 \u003d 0, which is impossible, to that in case of any kind of zero b, the value of the virus b 2 is 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, for the purpose of the square root, the equality b 2 \u003d a i c 2 \u003d a is valid, from them it is clear that 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, yet another square root at the warehouse a, then by the mirroring, similar to the one we have already pointed out, 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 the vislovlyuvannya.
For the subbags of the square root, the value of the square root is brought up, which is most 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’s only easy to get out of the understanding of the cube of the number, but not the square.
Appointment
Cubic root at warehouse 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 їx y cube: 7 3 \u003d 7 7 7 \u003d 343, 0 3 \u003d 0 0 0 \u003d 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, zavzhd 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.
Above those, there is only one cubic root from the whole number a. We bring the rest of the firmness. For which one, three vipadas are considered: 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 for a can be neither a negative number nor zero. True, let b є a cubic root for a, then for the same we can write equality b 3 \u003d a. Apparently, the confidence can be correct with negative b і for b=0 , the shards in the negatives b 3 =b·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, we can b=c, 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. Obviously, if you assume that the number b is used, if you see zero as a cube root from zero, then you can win evenness b 3 =0, as you can only for b=0.
For negative a, you can induce a mirroring, similar to the positive a. First, it is shown that the cubic 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 wine is obov'yazkovo with the first.
Otzzhe, zavzhd іsnuіє korіnіch s of any given decimal number a, moreover, one.
Damo designation of the arithmetic cube root.
Appointment
Arithmetic cube root of negative number a a number is called unknown to me, a cube of some old a.
The arithmetic cube root of an unknown number a is designated as , the sign is called the sign of the arithmetic cube root, the number 3 is called in this record 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 score the entries, for which the sign of the arithmetic cube root changes the negative numbers. Think of them 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 variation of the cube root, the reason is taken from the article of the variation of the roots: methods, applications, 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 some kind of a.
From the first appointment, it was understood that the root of the first step from the number a is the number a itself, the shards of the same step with the natural indicator took a 1 = a.
We have looked more closely at the n-th degree roots with n=2 and n=3 – the square root and the cube root. So the square root is the root of another level, and the cubic root is the root of the third level. For the extraction of the roots of the n-th step with n=4, 5, 6, ... they are manually divided into two groups: the first group - the roots of the paired steps (tobto, with n = 4, 6, 8, ...), the other group - 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 root, the steps of which are the guys of the number 4, 6, 8, ... As we already said, the stench is similar to the square root of the number a. Tobto, the root of any paired step in the warehouse a іsnuє only for non-negative 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 2m −c 2m =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 vіplivaє, scho b−c=0 аbo b+c=0 , аbo 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 when b is 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. Tobto, the root of the unpaired world from the number a is valid for the actual number a, moreover, for a given number a is one.
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 positive or offensive negative їх tvіr є positive number, then viraz b 2 +c 2 +b·c, which is in the arches 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, like the 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 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 a non-negative number is called, the n-th step of which is more a.
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 that come out, do not seem to get along? 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!
Podil root
We have risen from 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 sho, 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!
Upevnena, scho z usim, usim rested, 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 the warehouse - the same number, the square root of some old one.
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 how will the root be different? 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 of the world has a pair, but what if it will be unpaired? Well, I know, stop the level of power and spread everything into multipliers:
Z tsim nachebto everything is clear, but how to win the root of the z-pomіzh step? Axis, for example:
Easy to drink, right? And what is the bigger step for two? 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! I just got tired of trying to enter the number pid sign of the root!
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!)
Otrimanі vіdpovіdі it is necessary to roztashuvat on the coordinate line, for example, to determine which interval is suitable for the alignment. The first axis here is a clue: there is no calculator in use, but without it, how to reveal, which number is larger, and which is smaller? Axis and 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 root of great numbers
Before whom did we add the multiplier under the sign of the root, how can I blame it? It is necessary to simply lay out yoga on multipliers and win those who climb!
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! Laying out the leather multiplier under the roots on the okremi multipliers:
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 (guess signs of falsity):
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 a non-negative number is such a non-negative number, the square of which is better.
. - If we just take the square root of anything, then we always take one negative 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!