Chapter first.

Finding the largest integer square root from a given integer.

170. Preliminary remarks.

A) Since we will talk about extracting only the square root, to shorten the speech in this chapter, instead of “square” root we will say simply “root”.

b) If we square the numbers of the natural series: 1,2,3,4,5. . . , then we get the following table of squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100,121,144. .,

Obviously, there are a lot of integers that are not in this table; Of course, it is impossible to extract the whole root from such numbers. Therefore, if you need to extract the root of any integer, for example. required to find √4082, then we agree to understand this requirement as follows: extract the whole root of 4082, if possible; if it is not possible, then we must find the largest integer whose square is 4082 (such a number is 63, since 63 2 = 3969, and 64 2 = 4090).

V) If this number is less than 100, then the root of it is found using the multiplication table; Thus, √60 would be 7, since seven 7 equals 49, which is less than 60, and eight 8 equals 64, which is greater than 60.

171. Extracting the root of a number less than 10,000 but greater than 100. Let's say we need to find √4082. Since this number is less than 10,000, its root is less than √l0,000 = 100. On the other hand, this number is greater than 100; this means that the root of it is greater than (or equal to 10). (If, for example, it was necessary to find √ 120 , then although the number 120 > 100, however √ 120 is equal to 10, because 11 2 = 121.) But every number that is greater than 10 but less than 100 has 2 digits; This means that the required root is the sum:

tens + ones,

and therefore its square must equal the sum:

This sum must be the greatest square of 4082.

Let's take the largest of them, 36, and assume that the square of the tens root will be equal to exactly this largest square. Then the number of tens in the root must be 6. Let us now check that this should always be the case, i.e., the number of tens in the root is always equal to the largest integer root of the number of hundreds of the radical.

Indeed, in our example, the number of tens of the root cannot be more than 6, since (7 dec.) 2 = 49 hundreds, which exceeds 4082. But it cannot be less than 6, since 5 dec. (with units) is less than 6 des., and meanwhile (6 des.) 2 = 36 hundreds, which is less than 4082. And since we are looking for the largest whole root, we should not take 5 des for the root, when even 6 tens are not a lot of.

So, we have found the number of tens of the root, namely 6. We write this number to the right of the = sign, remembering that it means tens of the root. Raising it by the square, we get 36 hundreds. We subtract these 36 hundreds from the 40 hundreds of the radical number and subtract the remaining two digits of this number. The remainder 482 must contain 2 (6 dec.) (units) + (units)2. The product (6 dec.) (units) must be tens; therefore, the double product of tens by ones must be sought in the tens of the remainder, i.e., in 48 (we get their number by separating one digit on the right in the remainder of 48 "2). The doubled tens of the root make up 12. This means that if we multiply 12 by the units of the root ( which are still unknown), then we should get the number contained in 48. Therefore, we divide 48 by 12.

To do this, draw a vertical line to the left of the remainder and behind it (stepping back from the line one place to the left for the purpose that will now appear) we write double the first digit of the root, i.e. 12, and divide 48 by it. In the quotient we get 4.

However, we cannot guarantee in advance that the number 4 can be taken as units of the root, since we have now divided by 12 the entire number of tens of the remainder, while some of them may not belong to the double product of tens by units, but are part of the square of units. Therefore, the number 4 may be large. We need to try it out. It is obviously suitable if the sum 2 (6 dec.) 4 + 4 2 is no more than the remainder 482.

As a result, we get the sum of both at once. The resulting product turned out to be 496, which is greater than the remainder 482; That means number 4 is big. Then let's test the next smaller number 3 in the same way.

Examples.

In example 4, when dividing the 47 tens of the remainder by 4, we get 11 as a quotient. But since the number of units of the root cannot be a two-digit number 11 or 10, we must directly test the number 9.

In example 5, after subtracting 8 from the first face of the square, the remainder turns out to be 0, and the next face also consists of zeros. This shows that the desired root consists of only 8 tens, and therefore a zero must be put in place of the ones.

172. Extracting the root of a number greater than 10000. Let's say we need to find √35782. Since the radical number exceeds 10,000, the root of it is greater than √10000 = 100 and, therefore, it consists of 3 digits or more. No matter how many digits it consists of, we can always consider it as the sum of only tens and ones. If, for example, the root turns out to be 482, then we can count it as the amount of 48 des. + 2 units Then the square of the root will consist of 3 terms:

(dec.) 2 + 2 (dec.) (unit) + (unit) 2 .

Now we can reason in exactly the same way as when finding √4082 (in the previous paragraph). The only difference will be that to find the tens of the root of 4082 we had to extract the root of 40, and this could be done using the multiplication table; now, to obtain tens√35782, we will have to take the root of 357, which cannot be done using the multiplication table. But we can find √357 using the technique that was described in the previous paragraph, since the number 357< 10 000. Наибольший целый корень из 357 оказывается 18. Значит, в √3"57"82 должно быть 18 десятков. Чтобы найти единицы, надо из 3"57"82 вычесть квадрат 18 десятков, для чего достаточно вычесть квадрат 18 из 357 сотен и к остатку снести 2 последние цифры подкоренного числа. Остаток от вычитания квадpaта 18 из 357 у нас уже есть: это 33. Значит, для получения остатка от вычитания квадрата 18 дес. из 3"57"82, достаточно к 33 приписать справа цифры 82.

Next, we proceed as we did when finding √4082, namely: to the left of the remainder 3382 we draw a vertical line and behind it we write (stepping back one space from the line) twice the number of tens of the root found, i.e. 36 (twice 18). In the remainder, we separate one digit on the right and divide the number of tens of the remainder, i.e. 338, by 36. In the quotient we get 9. We test this number, for which we assign it to 36 on the right and multiply by it. The product turned out to be 3321, which is less than the remainder. This means that the number 9 is suitable, we write it at the root.

In general, to extract the square root of any integer, you must first extract the root of its hundreds; if this number is more than 100, then you will have to look for the root of the number of hundreds of these hundreds, that is, of the tens of thousands of this number; if this number is more than 100, you will have to take the root from the number of hundreds of tens of thousands, that is, from the millions of a given number, etc.

Examples.

In the last example, having found the first digit and subtracted its square, we get a remainder of 0. We subtract the next 2 digits 51. Separating the tens, we get 5 des, while the double found digit of the root is 6. This means that from dividing 5 by 6 we get 0 We put 0 in second place at the root and add the next 2 digits to the remainder; we get 5110. Then we continue as usual.

In this example, the required root consists of only 9 hundreds, and therefore zeros must be placed in the places of tens and in places of ones.

Rule. To extract the square root of a given integer, divide it from right hand to the left, on the edge, 2 digits each, except the last one, which may contain one digit.
To find the first digit of the root, take the square root of the first face.
To find the second digit, the square of the first digit of the root is subtracted from the first face, the second face is taken to the remainder, and the number of tens of the resulting number is divided by double the first digit of the root; the resulting integer is tested.
This test is carried out like this: behind the vertical line (to the left of the remainder) write twice the previously found number of the root and to it, with right side, the tested digit is assigned, the resulting number is multiplied by the tested digit after this addition. If after multiplication the result is a number greater than the remainder, then the tested digit is not suitable and the next smaller digit must be tested.
The next digits of the root are found using the same technique.
If, after demolishing the face, the number of tens of the resulting number turns out to be less than divisor, i.e. less than twice the found part of the root, then they put 0 at the root, remove the next face and continue the action further.

173. Number of digits of the root. From the consideration of the process of finding the root, it follows that there are as many digits in the root as there are faces of 2 digits each in the radical number (the left face may have one digit).

Chapter two.

Extracting confidants square roots from whole and fractional numbers .

For extracting the square root of polynomials, see the additions to the 2nd part of § 399 et seq.

174. Signs of an exact square root. The exact square root of a given number is a number whose square is exactly equal to the given number. Let us indicate some signs by which one can judge whether an exact root can be extracted from a given number or not:

A) If the exact whole root is not extracted from a given integer (the remainder is obtained when extracting), then the fractional exact root cannot be found from such a number, since any fraction that is not equal to a whole number, when multiplied by itself, also produces a fraction in the product, not an integer.

b) Since the root of a fraction is equal to the root of the numerator divided by the root of the denominator, the exact root of an irreducible fraction cannot be found if it cannot be extracted from the numerator or the denominator. For example, the exact root cannot be extracted from the fractions 4/5, 8/9 and 11/15, since in the first fraction it cannot be extracted from the denominator, in the second - from the numerator, and in the third - neither from the numerator nor from the denominator.

From numbers from which the exact root cannot be extracted, only approximate roots can be extracted.

175. Approximate root accurate to 1. An approximate square root, accurate to within 1, of a given number (integer or fractional, it doesn’t matter) is an integer that satisfies the following two requirements:

1) the square of this number is not greater than the given number; 2) but the square of this number increased by 1 is greater than this number. In other words, an approximate square root accurate to 1 is the largest integer square root of a given number, that is, the root that we learned to find in the previous chapter. This root is called approximate with an accuracy of 1, because to obtain an exact root, we would have to add some fraction less than 1 to this approximate root, so if instead of the unknown exact root we take this approximate one, we will make an error less than 1.

Rule. To extract an approximate square root accurate to within 1, you need to extract the largest integer root of the integer part of the given number.

The number found by this rule is an approximate root with a disadvantage , since it lacks the exact root of a certain fraction (less than 1). If we increase this root by 1, we get another number in which there is some excess over the exact root, and this excess is less than 1. This root increased by 1 can also be called an approximate root with an accuracy of 1, but with an excess. (The names: “with deficiency” or “with excess” in some mathematical books are replaced by other equivalent ones: “by deficiency” or “by excess.”)

176. Approximate root with an accuracy of 1/10. Let's say we need to find √2.35104 with an accuracy of 1/10. This means that you need to find a decimal fraction that would consist of whole units and tenths and that would satisfy the following two requirements:

1) the square of this fraction does not exceed 2.35104, but 2) if we increase it by 1/10, then the square of this increased fraction exceeds 2.35104.

To find such a fraction, we first find an approximate root accurate to 1, that is, we extract the root only from the integer 2. We get 1 (and the remainder is 1). We write the number 1 at the root and put a comma after it. Now we will look for the number of tenths. To do this, we take down to remainder 1 the digits 35 to the right of the decimal point, and continue extraction as if we were extracting the root of the integer 235. We write the resulting number 5 in the root in the place of tenths. We don't need the remaining digits of the radical number (104). That the resulting number 1.5 will actually be an approximate root with an accuracy of 1/10 can be seen from the following. If we were to find the largest integer root of 235 with an accuracy of 1, we would get 15. So:

15 2 < 235, but 16 2 >235.

Dividing all these numbers by 100, we get:

This means that the number 1.5 is the decimal fraction that we called an approximate root with an accuracy of 1/10.

Using this technique, we can also find the following approximate roots with an accuracy of 0.1:

177. Approximate square root to within 1/100 to 1/1000, etc.

Suppose we need to find an approximate √248 with an accuracy of 1/100. This means: find a decimal fraction that would consist of whole, tenths and hundredths parts and that would satisfy two requirements:

1) its square does not exceed 248, but 2) if we increase this fraction by 1/100, then the square of this increased fraction exceeds 248.

We will find such a fraction in the following sequence: first we will find the whole number, then the tenths figure, then the hundredths figure. The root of an integer is 15 integers. To get the tenths figure, as we have seen, you need to add to the remainder 23 2 more digits to the right of the decimal point. In our example, these numbers are not present at all; we put zeros in their place. By adding them to the remainder and continuing as if we were finding the root of the integer 24,800, we will find the tenths figure 7. It remains to find the hundredths figure. To do this, we add 2 more zeros to the remainder 151 and continue extraction, as if we were finding the root of the integer 2,480,000. We get 15.74. That this number is really an approximate root of 248 with an accuracy of 1/100 can be seen from the following. If we were to find the largest integer square root of the integer 2,480,000, we would get 1574; Means:

1574 2 < 2,480,000, but 1575 2 > 2,480,000.

Dividing all numbers by 10,000 (= 100 2), we get:

This means that 15.74 is that decimal fraction that we called an approximate root with an accuracy of 1/100 of 248.

Applying this technique to finding an approximate root with an accuracy of 1/1000 to 1/10000, etc., we find the following.

Rule. To extract from this whole numbers or from a given decimal fraction an approximate root with an accuracy of 1/10 to 1/100 to 1/100, etc., first find an approximate root with an accuracy of 1, extracting the root from the integer (if it is not there, write about the root 0 whole).

Then they find the number of tenths. To do this, add to the remainder the 2 digits of the radical number to the right of the decimal point (if they are not there, add two zeros to the remainder), and continue extraction as is done when extracting the root of an integer. The resulting number is written at the root in the place of tenths.

Then find the hundredths number. To do this, two numbers to the right of those that were just removed are added to the remainder, etc.

Thus, when extracting the root of an integer with a decimal fraction, it is necessary to divide into faces 2 digits each, starting from the decimal point, both to the left (in the integer part of the number) and to the right (in the fractional part).

Examples.

1) Find up to 1/100 roots: a) √2; b) √0.3;

In the last example, we converted the fraction 3/7 to a decimal by calculating 8 decimal places to form the 4 faces needed to find the 4 decimal places of the root.

178. Description of the table of square roots. At the end of this book is a table of square roots calculated with four digits. Using this table, you can quickly find the square root of a whole number (or decimal fraction) that is expressed in no more than four digits. Before explaining how this table is structured, we note that we can always find the first significant digit of the desired root without the help of tables by just looking at the radical number; we can also easily determine which decimal place the first digit of the root means and, therefore, where in the root, when we find its digits, we must put a comma. Here are some examples:

1) √5"27,3 . The first digit will be 2, since the left side of the radical number is 5; and the root of 5 is equal to 2. In addition, since in the integer part of the radical there are only 2 faces, then in the integer part of the desired root there must be 2 digits and, therefore, its first digit 2 must mean tens.

2) √9.041. Obviously, in this root the first digit will be 3 prime units.

3) √0.00"83"4. The first significant digit is 9, since the face from which the root would have to be taken to obtain the first significant digit is 83, and the root of 83 is 9. Since the required number will not contain either whole numbers or tenths, the first digit 9 must mean hundredths.

4) √0.73"85. The first significant figure is 8 tenths.

5) √0.00"00"35"7. The first significant figure will be 5 thousandths.

Let's make one more remark. Let us assume that we need to extract the root of a number which, after discarding the occupied word in it, is represented by a series of numbers like this: 5681. This root can be one of the following:

If we take the roots that we underline with one line, then they will all be expressed by the same series of numbers, precisely those numbers that are obtained when extracting the root from 5681 (these will be the numbers 7, 5, 3, 7). The reason for this is that the faces into which the radical number has to be divided when finding the digits of the root will be the same in all these examples, therefore the digits for each root will be the same (only the position of the decimal point will, of course, be different). In the same way, in all the roots underlined by us with two lines, we should get same numbers, precisely those by which √568.1 is expressed (these numbers will be 2, 3, 8, 3), and for the same reason. Thus, the digits of the roots of the numbers represented (by dropping the comma) by the same row of numbers 5681 will be of two (and only two) kind: either this is the row 7, 5, 3, 7, or the row 2, 3, 8, 3. The same, obviously, can be said about any other series of numbers. Therefore, as we will now see, in the table, each row of digits of the radical number corresponds to 2 rows of digits for the roots.

Now we can explain the structure of the table and how to use it. For clarity of explanation, we have shown the beginning of the first page of the table here.

This table is located on several pages. On each of them, in the first column on the left, the numbers 10, 11, 12... (up to 99) are placed. These numbers express the first 2 digits of the number from which the square root is sought. In the top horizontal line (as well as in the bottom) are the numbers: 0, 1, 2, 3... 9, representing the 3rd digit of this number, and then further to the right are the numbers 1, 2, 3. . . 9, representing the 4th digit of this number. All other horizontal lines contain 2 four-digit numbers expressing the square roots of the corresponding numbers.

Suppose you need to find the square root of some number, integer or expressed decimal. First of all, we find, without the help of tables, the first digit of the root and its digit. Then we will discard the comma in this number, if there is one. Let us first assume that after discarding the comma, only 3 digits will remain, for example. 114. We find in the tables in the leftmost column the first 2 digits, i.e. 11, and move from them to the right along the horizontal line until we reach the vertical column, at the top (and bottom) of which is the 3rd digit of the number , i.e. 4. In this place we find two four-digit numbers: 1068 and 3376. Which of these two numbers should be taken and where to place the comma in it, this is determined by the first digit of the root and its digit, which we found earlier. So, if we need to find √0.11"4, then the first digit of the root is 3 tenths, and therefore we must take 0.3376 for the root. If we needed to find √1.14, then the first digit of the root would be 1, and we Then we would take 1.068.

This way we can easily find:

√5.30 = 2.302; √7"18 = 26.80; √0.91"6 = 0.9571, etc.

Let us now assume that we need to find the root of a number expressed (by dropping the decimal point) in 4 digits, for example, √7"45.6. Noting that the first digit of the root is 2 tens, we find for the number 745, as has now been explained, the digits 2729 (we only notice this number with our finger, but do not write it down.) Then we move from this number further to the right until on the right side of the table (behind the last bold line) we meet the vertical column that is marked at the top (and bottom) 4 the th digit of the given number, i.e. the number 6, and find the number 1 there. This will be a correction that must be applied (in the mind) to the previously found number 2729; we get 2730. We write this number down and put a comma in it in the proper place : 27.30.

In this way we find, for example:

√44.37 = 6.661; √4.437 = 2.107; √0.04"437 =0.2107, etc.

If the radical number is expressed by only one or two digits, then we can assume that these digits are followed by one or two zeros, and then proceed as explained for a three-digit number. For example, √2.7 =√2.70 =1.643; √0.13 = √0.13"0 = 0.3606, etc..

Finally, if the radical number is expressed by more than 4 digits, then we will take only the first 4 of them, and discard the rest, and to reduce the error, if the first of the discarded digits is 5 or more than 5, then we will increase by l the fourth of the retained digits . So:

√357,8| 3 | = 18,91; √0,49"35|7 | = 0.7025; and so on.

Comment. The tables indicate the approximate square root, sometimes with a deficiency, sometimes with an excess, namely the one of these approximate roots that comes closer to the exact root.

179. Extracting square roots from ordinary fractions. The exact square root of an irreducible fraction can be extracted only when both terms of the fraction are exact squares. In this case, it is enough to extract the root of the numerator and denominator separately, for example:

The approximate square root of an ordinary fraction with some decimal precision can most easily be found if we first reverse common fraction to a decimal, calculating in this fraction the number of decimal places after the decimal point that would be twice the number of decimal places in the desired root.

However, you can do it differently. Let's explain this with the following example:

Find approximate √ 5 / 24

Let's make the denominator an exact square. To do this, it would be enough to multiply both terms of the fraction by the denominator 24; but in this example you can do it differently. Let's decompose 24 into prime factors: 24 = 2 2 2 3. From this decomposition it is clear that if 24 is multiplied by 2 and another 3, then each prime factor will be repeated in the product even number times, and therefore the denominator becomes a square:

It remains to calculate √30 with some accuracy and divide the result by 12. It must be borne in mind that dividing by 12 will also reduce the fraction indicating the degree of accuracy. So, if we find √30 with an accuracy of 1/10 and divide the result by 12, we will obtain an approximate root of the fraction 5/24 with an accuracy of 1/120 (namely 54/120 and 55/120)

Chapter three.

Graph of a functionx = √y .

180. Inverse function. Let some equation be given that determines at as a function of X , for example, like this: y = x 2 . We can say that it determines not only at as a function of X , but also, conversely, determines X as a function of at , albeit in an implicit way. To make this function explicit, we need to solve given equation relatively X , taking at for a known number; So, from the equation we took we find: y = x 2 .

The algebraic expression obtained for x after solving the equation that defines y as a function of x is called the inverse function of the one that defines y.

So the function x = √y inverse function y = x 2 . If, as is customary, we denote the independent variable X , and the dependent at , then the inverse function obtained now can be expressed as follows: y = √x . Thus, in order to obtain a function inverse to a given (direct) one, it is necessary to derive from the equation defining this given function X depending on the y and in the resulting expression replace y on x , A X on y .

181. Graph of a function y = √x . This function is not possible with a negative value X , but it is possible to calculate it (with any accuracy) for any positive value x , and for each such value the function receives two different meanings with the same absolute value, nose opposite signs. If you are familiar √ If we denote only the arithmetic value of the square root, then these two values ​​of the function can be expressed as follows: y = ± √x To plot a graph of this function, you must first compile a table of its values. The easiest way to create this table is from the table of direct function values:

y = x 2 .

x

y

if the values at take as values X , and vice versa:

y = ± √x

By plotting all these values ​​on the drawing, we get the following graph.

In the same drawing we depicted (with a broken line) the graph of the direct function y = x 2 . Let's compare these two graphs with each other.

182. The relationship between the graphs of direct and inverse functions. To compile a table of values ​​of the inverse function y = ± √x we took for X those numbers that are in the table of the direct function y = x 2 served as values ​​for at , and for at took those numbers; which in this table were the values ​​for x . It follows from this that both graphs are the same, only the graph of the direct function is so located relative to the axis at - how the graph of the inverse function is located relative to the axis X - ov. As a result, if we bend the drawing around a straight line OA bisecting a right angle xOy , so that the part of the drawing containing the semi-axis OU , fell on the part that contains the axle shaft Oh , That OU compatible with Oh , all divisions OU will coincide with divisions Oh , and parabola points y = x 2 will align with the corresponding points on the graph y = ± √x . For example, points M And N , whose ordinate 4 , and the abscissas 2 And - 2 , will coincide with the points M" And N" , for which the abscissa 4 , and the ordinates 2 And - 2 . If these points coincide, this means that the straight lines MM" And NN" perpendicular to OA and divide this straight line in half. The same can be said for all other corresponding points in both graphs.

Thus, the graph of the inverse function should be the same as the graph of the direct function, but these graphs are located differently, namely symmetrically with each other relative to the bisector of the angle xOy . We can say that the graph of the inverse function is a reflection (as in a mirror) of the graph of the direct function relative to the bisector of the angle xOy .

What is a square root?

Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)

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 any number without special 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 one is complicated 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...

For free and successful work 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 uncertainty to own strength... Let's deal with this issue 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 does 4 square? 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 are positive. Even a special term was invented: from the number A- This non-negative number whose square is A. 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 (just 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 decide something quadratic equation, type:

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.

Extracting the root is the reverse operation of raising a power. That is, taking the root of the number X, we get a number that squared will give the same number X.

Extracting the root is a fairly simple operation. A table of squares can make the extraction work easier. Because it is impossible to remember all the squares and roots by heart, but the numbers may be large.

Extracting the root of a number

Taking the square root of a number is easy. Moreover, this can be done not immediately, but gradually. For example, take the expression √256. Initially, it is difficult for an ignorant person to give an answer right away. Then we will do it step by step. First, we divide by just the number 4, from which we take the selected square as the root.

Let's represent: √(64 4), then it will be equivalent to 2√64. And as you know, according to the multiplication table 64 = 8 8. The answer will be 2*8=16.

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Extracting a complex root

The square root cannot be calculated from negative numbers, because any squared number is a positive number!

A complex number is the number i, which squared is equal to -1. That is, i2=-1.

In mathematics, there is a number that is obtained by taking the root of the number -1.

That is, it is possible to calculate the root of a negative number, but this already applies to higher mathematics, not school mathematics.

Let's consider an example of such a root extraction: √(-49)=7*√(-1)=7i.

Online root calculator

Using our calculator, you can calculate the extraction of a number from the square root:

Converting Expressions Containing a Root Operation

The essence of transforming radical expressions is to decompose the radical number into simpler ones, from which the root can be extracted. Such as 4, 9, 25 and so on.

Let's give an example, √625. Let's divide the radical expression by the number 5. We get √(125 5), repeat the operation √(25 25), but we know that 25 is 52. Which means the answer will be 5*5=25.

But there are numbers for which the root cannot be calculated using this method and you just need to know the answer or have a table of squares at hand.

√289=√(17*17)=17

Bottom line

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From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special tasks and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.

And do you have calculator addiction? Or do you think that it is very difficult to calculate, for example, except with a calculator or using a table of squares.

It happens that schoolchildren are tied to a calculator and even multiply 0.7 by 0.5 by pressing the treasured buttons. They say, well, I still know how to calculate, but now I’ll save time... When the exam comes... then I’ll strain myself...

So the fact is that there will already be plenty of “stressful moments” during the exam... As they say, water wears away stones. So in an exam, little things, if there are a lot of them, can ruin you...

Let's minimize the number of possible troubles.

Taking the square root of a large number

We will now only talk about the case when the result of extracting the square root is an integer.

Case 1.

So, let us at any cost (for example, when calculating the discriminant) need to calculate the square root of 86436.

We will factor the number 86436 into prime factors. Divide by 2, we get 43218; divide by 2 again, we get 21609. A number cannot be divisible by 2. But since the sum of the digits is divisible by 3, then the number itself is divisible by 3 (generally speaking, it is clear that it is also divisible by 9). . Divide by 3 again, and we get 2401. 2401 is not completely divisible by 3. Not divisible by five (does not end in 0 or 5).

We suspect divisibility by 7. Indeed, and ,

So, Complete order!

Case 2.

Let us need to calculate . It is inconvenient to act in the same way as described above. We are trying to factorize...

The number 1849 is not divisible by 2 (it is not even)…

It is not completely divisible by 3 (the sum of the digits is not a multiple of 3)...

It is not completely divisible by 5 (the last digit is neither 5 nor 0)…

It’s not completely divisible by 7, it’s not divisible by 11, it’s not divisible by 13... Well, how long will it take us to sort through all the prime numbers?

Let's think a little differently.

We understand that

We have narrowed our search. Now we go through the numbers from 41 to 49. Moreover, it is clear that since the last digit of the number is 9, then we should stop at options 43 or 47 - only these numbers, when squared, will give the last digit 9.

Well, here, of course, we stop at 43. Indeed,

P.S. How the hell do we multiply 0.7 by 0.5?

You should multiply 5 by 7, ignoring the zeros and signs, and then separate, going from right to left, two decimal places. We get 0.35.

Instructions

Select a multiplier for the radical number, the removal of which from under root is really an expression - otherwise the operation will lose . For example, if under the sign root with an exponent equal to three (cube root), it costs number 128, then from under the sign you can take out, for example, number 5. At the same time, the radical number 128 will have to be divided by 5 cubed: ³√128 = 5∗³√(128/5³) = 5∗³√(128/125) = 5∗³√1.024. If availability fractional number under the sign root does not contradict the conditions of the problem, then it is possible in this form. If you need a simpler option, then first break the radical expression into such integer factors, the cube root of one of which will be an integer number m. For example: ³√128 = ³√(64∗2) = ³√(4³∗2) = 4∗³√2.

Use to select factors of a radical number if it is not possible to calculate the powers of a number in your head. This is especially true for root m with an exponent greater than two. If you have access to the Internet, you can perform calculations using the calculators built into the Google and Nigma search engines. For example, if you need to find the largest integer factor that can be taken out from under the cubic sign root for the number 250, then go to the Google website and enter the query “6^3” to check if it is possible to remove it from under the sign root six. The search engine will show a result equal to 216. Alas, 250 cannot be divided without a remainder by this number. Then enter the query 5^3. The result will be 125, and this allows you to divide 250 into factors of 125 and 2, which means taking it out of the sign root number 5, leaving there number 2.

Sources:

• how to get it out from under the roots
• Square root of the product

Take it out from under root one of the factors is necessary in situations where you need to simplify a mathematical expression. There are times when it is impossible to perform the necessary calculations using a calculator. For example, if letter designations for variables are used instead of numbers.

Instructions

Break down the radical expression into simple factors. See which of the factors is repeated the same number of times, indicated in the indicators root, or more. For example, you need to take the fourth root of a. In this case, the number can be represented as a*a*a*a = a*(a*a*a)=a*a3. Indicator root in this case it will correspond with factor a3. It needs to be taken out of the sign.

Extract the root of the resulting radicals separately where possible. Extraction root is the algebraic operation inverse to exponentiation. Extraction root of an arbitrary power, find a number from a number that, when raised to this arbitrary power, will result in the given number. If extraction root cannot be produced, leave the radical expression under the sign root just the way it is. As a result of the above actions, you will be removed from under sign root.

Video on the topic

note

Be careful when writing radical expressions in the form of factors - an error at this stage will lead to incorrect results.

Helpful advice

When extracting roots, it is convenient to use special tables or tables of logarithmic roots - this will significantly reduce the time it takes to find the right decision.

Sources:

• root extraction sign in 2019

Simplification of algebraic expressions is required in many areas of mathematics, including solving higher-order equations, differentiation and integration. Several methods are used, including factorization. To apply this method, you need to find and make a general factor behind brackets.

Instructions

Carrying out the total multiplier brackets- one of the most common methods of decomposition. This technique is used to simplify the structure of long algebraic expressions, i.e. polynomials. The general number can be a number, a monomial or a binomial, and to find it, the distributive property of multiplication is used.

Number. Look carefully at the coefficients of each polynomial to see if they can be divided by the same number. For example, in the expression 12 z³ + 16 z² – 4 it is obvious factor 4. After the transformation, you get 4 (3 z³ + 4 z² - 1). In other words, this number is the least common integer divisor of all coefficients.

Monomial. Determine whether the same variable is in each of the terms of the polynomial. Assuming this is the case, now look at the coefficients as in the previous case. Example: 9 z^4 – 6 z³ + 15 z² – 3 z.

Each element of this polynomial contains a variable z. In addition, all coefficients are numbers that are multiples of 3. Therefore, the common factor will be the monomial 3 z:3 z (3 z³ – 2 z² + 5 z - 1).

Binomial.For brackets general factor of two, a variable and a number, which is a common polynomial. Therefore, if factor-the binomial is not obvious, then you need to find at least one root. Select the free term of the polynomial; this is a coefficient without a variable. Now apply the method of substitution into the general expression of all integer divisors of the free term.

Consider: z^4 – 2 z³ + z² - 4 z + 4. Check to see if any of the integer factors of 4 are z^4 – 2 z³ + z² - 4 z + 4 = 0. By simple substitution, find z1 = 1 and z2 = 2, which means for brackets we can remove the binomials (z - 1) and (z - 2). To find the remaining expression, use sequential long division.