It's time to disassemble root extraction methods. They are based on the properties of the roots, in particular, on the equality, which is true for any non-negative number b.

Below we will consider in turn the main methods of extracting roots.

Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.

If the tables of squares, cubes, etc. is not at hand, it is logical to use the method of extracting the root, which involves decomposing the root number into simple factors.

Separately, it is worth dwelling on, which is possible for roots with odd exponents.

Finally, consider a method that allows you to sequentially find the digits of the value of the root.

Let's get started.

Using a table of squares, a table of cubes, etc.

In the simplest cases, tables of squares, cubes, etc. allow extracting roots. What are these tables?

The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a certain row and a certain column, it allows you to make a number from 0 to 99. For example, let's select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each of its cells is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99 . At the intersection of our chosen row of 8 tens and column 3 of one, there is a cell with the number 6889, which is the square of the number 83.

Tables of cubes, tables of fourth powers of numbers from 0 to 99 and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.

Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. respectively from the numbers in these tables. Let us explain the principle of their application in extracting roots.

Let's say we need to extract the n-th root of the number a, while the number a is contained in the table of n-th degrees. According to this table, we find the number b such that a=b n . Then , therefore, the number b will be the desired root of the nth degree.

As an example, let's show how the cube root of 19683 is extracted using the cube table. We find the number 19 683 in the table of cubes, from it we find that this number is a cube of the number 27, therefore, .

It is clear that tables of n-th degrees are very convenient when extracting roots. However, they are often not at hand, and their compilation requires a certain amount of time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, one has to resort to other methods of extracting the roots.

Decomposition of the root number into prime factors

A fairly convenient way to extract the root from a natural number (if, of course, the root is extracted) is to decompose the root number into prime factors. His the essence is as follows: after it is quite easy to represent it as a degree with the desired indicator, which allows you to get the value of the root. Let's explain this point.

Let the root of the nth degree be extracted from a natural number a, and its value is equal to b. In this case, the equality a=b n is true. The number b as any natural number can be represented as a product of all its prime factors p 1 , p 2 , …, p m in the form p 1 p 2 … p m , and the root number a in this case is represented as (p 1 p 2 ... p m) n . Since the decomposition of the number into prime factors is unique, the decomposition of the root number a into prime factors will look like (p 1 ·p 2 ·…·p m) n , which makes it possible to calculate the value of the root as .

Note that if the factorization of the root number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n , then the root of the nth degree from such a number a is not completely extracted.

Let's deal with this when solving examples.

Example.

Take the square root of 144 .

Solution.

If we turn to the table of squares given in the previous paragraph, it is clearly seen that 144=12 2 , from which it is clear that the square root of 144 is 12 .

But in the light of this point, we are interested in how the root is extracted by decomposing the root number 144 into prime factors. Let's take a look at this solution.

Let's decompose 144 to prime factors:

That is, 144=2 2 2 2 3 3 . Based on the resulting decomposition, the following transformations can be carried out: 144=2 2 2 2 3 3=(2 2) 2 3 2 =(2 2 3) 2 =12 2. Consequently, .

Using the properties of the degree and properties of the roots, the solution could be formulated a little differently: .

Answer:

To consolidate the material, consider the solutions of two more examples.

Example.

Calculate the root value.

Solution.

The prime factorization of the root number 243 is 243=3 5 . In this way, .

Answer:

Example.

Is the value of the root an integer?

Solution.

To answer this question, let's decompose the root number into prime factors and see if it can be represented as a cube of an integer.

We have 285 768=2 3 3 6 7 2 . The resulting decomposition is not represented as a cube of an integer, since the degree of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 is not taken completely.

Answer:

No.

Extracting roots from fractional numbers

It's time to figure out how the root is extracted from a fractional number. Let the fractional root number be written as p/q . According to the property of the root of the quotient, the following equality is true. From this equality it follows fraction root rule: The root of a fraction is equal to the quotient of dividing the root of the numerator by the root of the denominator.

Let's look at an example of extracting a root from a fraction.

Example.

What is the square root of the common fraction 25/169.

Solution.

According to the table of squares, we find that the square root of the numerator of the original fraction is 5, and the square root of the denominator is 13. Then . This completes the extraction of the root from an ordinary fraction 25/169.

Answer:

The root of a decimal fraction or a mixed number is extracted after replacing the root numbers with ordinary fractions.

Example.

Take the cube root of the decimal 474.552.

Solution.

Let's represent the original decimal as a common fraction: 474.552=474552/1000 . Then . It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2 2 2 3 3 3 13 13 13=(2 3 13) 3 =78 3 and 1 000=10 3 , then and . It remains only to complete the calculations .

Answer:

.

Extracting the root of a negative number

Separately, it is worth dwelling on extracting roots from negative numbers. When studying roots, we said that when the exponent of the root is an odd number, then a negative number can be under the sign of the root. We gave such notations the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, we have . This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to extract the root of the opposite positive number, and put a minus sign in front of the result.

Let's consider an example solution.

Example.

Find the root value.

Solution.

Let's transform the original expression so that a positive number appears under the root sign: . Now we replace the mixed number with an ordinary fraction: . We apply the rule of extracting the root from an ordinary fraction: . It remains to calculate the roots in the numerator and denominator of the resulting fraction: .

Here is a summary of the solution: .

Answer:

.

Bitwise Finding the Root Value

In the general case, under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But at the same time, there is a need to know the value of a given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to consistently obtain a sufficient number of values ​​​​of the digits of the desired number.

The first step of this algorithm is to find out what is the most significant bit of the root value. To do this, the numbers 0, 10, 100, ... are successively raised to the power n until a number exceeding the root number is obtained. Then the number that we raised to the power of n in the previous step will indicate the corresponding high order.

For example, consider this step of the algorithm when extracting the square root of five. We take the numbers 0, 10, 100, ... and square them until we get a number greater than 5 . We have 0 2 =0<5 , 10 2 =100>5 , which means that the most significant digit will be the units digit. The value of this bit, as well as lower ones, will be found in the next steps of the root extraction algorithm.

All the following steps of the algorithm are aimed at successive refinement of the value of the root due to the fact that the values ​​of the next digits of the desired value of the root are found, starting from the highest and moving to the lowest. For example, the value of the root in the first step is 2 , in the second - 2.2 , in the third - 2.23 , and so on 2.236067977 ... . Let us describe how the values ​​of the bits are found.

Finding bits is carried out by enumeration of their possible values ​​0, 1, 2, ..., 9 . In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the root number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and the transition to the next step of the root extraction algorithm is made, if this does not happen, then the value of this digit is 9 .

Let us explain all these points using the same example of extracting the square root of five.

First, find the value of the units digit. We will iterate over the values ​​0, 1, 2, …, 9 , calculating respectively 0 2 , 1 2 , …, 9 2 until we get a value greater than the radical number 5 . All these calculations are conveniently presented in the form of a table:

So the value of the units digit is 2 (because 2 2<5 , а 2 3 >5 ). Let's move on to finding the value of the tenth place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the obtained values ​​\u200b\u200bwith the root number 5:

Since 2.2 2<5 , а 2,3 2 >5 , then the value of the tenth place is 2 . You can proceed to finding the value of the hundredths place:

So the next value of the root of five is found, it is equal to 2.23. And so you can continue to find values ​​further: 2,236, 2,2360, 2,23606, 2,236067, … .

To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.

First, we define the senior digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151.186 . We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151.186 , so the most significant digit is the tens digit.

Let's define its value.

Since 10 3<2 151,186 , а 20 3 >2,151.186 , then the value of the tens digit is 1 . Let's move on to units.

Thus, the value of the ones place is 2 . Let's move on to ten.

Since even 12.9 3 is less than the radical number 2 151.186 , the value of the tenth place is 9 . It remains to perform the last step of the algorithm, it will give us the value of the root with the required accuracy.

At this stage, the value of the root is found up to hundredths: .

In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, those that we studied above are sufficient.

Bibliography.

• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

Root formulas. properties of square roots.

Attention!
There are additional
material in Special Section 555.
For those who strongly "not very..."
And for those who "very much...")

In the previous lesson, we figured out what a square root is. It's time to figure out what are formulas for roots, what are root properties and what can be done about it all.

Root Formulas, Root Properties, and Rules for Actions with Roots- it's essentially the same thing. There are surprisingly few formulas for square roots. Which, of course, pleases! Rather, you can write a lot of all sorts of formulas, but only three are enough for practical and confident work with roots. Everything else flows from these three. Although many stray in the three formulas of the roots, yes ...

Let's start with the simplest. There she is:

If you like this site...

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

You can practice solving examples and find out your level. Testing with instant verification. Learning - with interest!)

you can get acquainted with functions and derivatives.

Among the many knowledge that is a sign of literacy, the alphabet is in the first place. The next, the same "sign" element, are the skills of addition-multiplication and, adjacent to them, but reverse in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, And everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to take roots in life, except in the country? For example, such an entertaining problem, like, the square root of the number 12345 ... Is there still gunpowder in the powder flasks? Can we do it? Yes, there is nothing easier! Where is my calculator ... And without it, hand-to-hand, weak?

First, let's clarify what it is - the square root of a number. Generally speaking, "to extract a root from a number" means to perform the arithmetic operation opposite to raising to a power - here you have the unity of opposites in life application. let's say a square is a multiplication of a number by itself, i.e., as they taught at school, X * X = A or in another notation X2 = A, and in words - “X squared equals A”. Then the inverse problem sounds like this: the square root of the number A, is the number X, which, when squared, is equal to A.

Extracting the square root

From the school course of arithmetic, methods of calculations "in a column" are known, which help to perform any calculations using the first four arithmetic operations. Alas ... For square, and not only square, roots of such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequential enumeration of numbers, the square of which approaches the value of the root expression. Only and everything! Before an hour or two has passed, it can be calculated using the well-known method of multiplying into a “column”, any square root. If you have the skills, a couple of minutes is enough for this. Even a not quite advanced calculator or PC user does it in one fell swoop - progress.

But seriously, the calculation of the square root is often performed using the “artillery fork” technique: first, they take a number whose square approximately corresponds to the root expression. It is better if "our square" is slightly less than this expression. Then they correct the number according to their own skill-understanding, for example, multiply by two, and ... square it again. If the result is greater than the number under the root, successively adjusting the original number, gradually approaching its "colleague" under the root. As you can see - no calculator, only the ability to count "in a column". Of course, there are many scientifically reasoned and optimized algorithms for calculating the square root, but for "home use" the above technique gives 100% confidence in the result.

Yes, I almost forgot, in order to confirm our increased literacy, we calculate the square root of the previously indicated number 12345. We do it step by step:

1. Take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is on top - the result is less than 12345.

2. Let's try, also purely intuitively, X = 120. Then: X * X = 14400. And again, with intuition, the order - the result is more than 12345.

3. Above, a “fork” of 100 and 120 is obtained. Let's choose new numbers - 110 and 115. We get, respectively, 12100 and 13225 - the fork narrows.

4. We try on "maybe" X = 111. We get X * X = 12321. This number is already quite close to 12345. In accordance with the required accuracy, the “fitting” can be continued or stopped at the result obtained. That's all. As promised - everything is very simple and without a calculator.

Quite a bit of history...

Even the Pythagoreans, students of the school and followers of Pythagoras, thought of using square roots, 800 BC. and right there, "ran into" new discoveries in the field of numbers. And where did it come from?

1. The solution of the problem with the extraction of the root, gives the result in the form of numbers of a new class. They were called irrational, in other words, "unreasonable", because. they are not written as a complete number. The most classic example of this kind is the square root of 2. This case corresponds to the calculation of the diagonal of a square with a side equal to 1 - here it is, the influence of the Pythagorean school. It turned out that in a triangle with a very specific unit size of the sides, the hypotenuse has a size that is expressed by a number that "has no end." So in mathematics appeared

2. It is known that It turned out that this mathematical operation contains one more trick - extracting the root, we do not know what square of which number, positive or negative, is the root expression. This uncertainty, the double result from one operation, is written down.

The study of the problems associated with this phenomenon has become a direction in mathematics called the theory of a complex variable, which is of great practical importance in mathematical physics.

It is curious that the designation of the root - radical - was used in his "Universal Arithmetic" by the same ubiquitous I. Newton, and exactly the modern form of writing the root has been known since 1690 from the Frenchman Roll's book "Algebra Manual".

The area of ​​a square plot of land is 81 dm². Find his side. Suppose the length of the side of the square is X decimetres. Then the area of ​​the plot is X² square decimetres. Since, according to the condition, this area is 81 dm², then X² = 81. The length of the side of a square is a positive number. A positive number whose square is 81 is the number 9. When solving the problem, it was required to find the number x, the square of which is 81, i.e. solve the equation X² = 81. This equation has two roots: x 1 = 9 and x 2 \u003d - 9, since 9² \u003d 81 and (- 9)² \u003d 81. Both numbers 9 and - 9 are called the square roots of the number 81.

Note that one of the square roots X= 9 is a positive number. It is called the arithmetic square root of 81 and is denoted √81, so √81 = 9.

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

For example, the numbers 6 and -6 are the square roots of 36. The number 6 is the arithmetic square root of 36, since 6 is a non-negative number and 6² = 36. The number -6 is not an arithmetic root.

Arithmetic square root of a number a denoted as follows: √ a.

The sign is called the arithmetic square root sign; a is called a root expression. Expression √ a read like this: the arithmetic square root of a number a. For example, √36 = 6, √0 = 0, √0.49 = 0.7. In cases where it is clear that we are talking about an arithmetic root, they briefly say: "the square root of a«.

The act of finding the square root of a number is called taking the square root. This action is the reverse of squaring.

Any number can be squared, but not every number can be square roots. For example, it is impossible to extract the square root of the number - 4. If such a root existed, then, denoting it with the letter X, we would get the wrong equality x² \u003d - 4, since there is a non-negative number on the left, and a negative one on the right.

Expression √ a only makes sense when a ≥ 0. The definition of the square root can be briefly written as: √ a ≥ 0, (√a)² = a. Equality (√ a)² = a valid for a ≥ 0. Thus, to make sure that the square root of a non-negative number a equals b, i.e., that √ a =b, you need to check that the following two conditions are met: b ≥ 0, b² = a.

The square root of a fraction

Let's calculate . Note that √25 = 5, √36 = 6, and check if the equality holds.

Because and , then the equality is true. So, .

Theorem: If a a≥ 0 and b> 0, that is, the root of the fraction is equal to the root of the numerator divided by the root of the denominator. It is required to prove that: and .

Since √ a≥0 and √ b> 0, then .

By the property of raising a fraction to a power and determining the square root the theorem is proven. Let's look at a few examples.

Calculate , according to the proven theorem .

Second example: Prove that , if a ≤ 0, b < 0. .

Another example: Calculate .

.

Square root transformation

Taking the multiplier out from under the sign of the root. Let an expression be given. If a a≥ 0 and b≥ 0, then by the theorem on the root of the product, we can write:

Such a transformation is called factoring out the root sign. Consider an example;

Calculate at X= 2. Direct substitution X= 2 in the radical expression leads to complicated calculations. These calculations can be simplified if we first remove the factors from under the root sign: . Now substituting x = 2, we get:.

So, when taking out the factor from under the root sign, the radical expression is represented as a product in which one or more factors are the squares of non-negative numbers. The root product theorem is then applied and the root of each factor is taken. Consider an example: Simplify the expression A = √8 + √18 - 4√2 by taking out the factors from under the root sign in the first two terms, we get:. We emphasize that the equality valid only when a≥ 0 and b≥ 0. if a < 0, то .

Quite often, when solving problems, we are faced with large numbers from which we need to extract Square root. Many students decide that this is a mistake and start resolving the whole example. Under no circumstances should this be done! There are two reasons for this:

1. The roots of large numbers do occur in problems. Especially in text;
2. There is an algorithm by which these roots are considered almost verbally.

We will consider this algorithm today. Perhaps some things will seem incomprehensible to you. But if you pay attention to this lesson, you will get the most powerful weapon against square roots.

So the algorithm:

1. Limit the desired root above and below to multiples of 10. Thus, we will reduce the search range to 10 numbers;
2. From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
3. Square these 1-2 numbers. That of them, the square of which is equal to the original number, will be the root.

Before applying this algorithm works in practice, let's look at each individual step.

Roots constraint

First of all, we need to find out between which numbers our root is located. It is highly desirable that the numbers be a multiple of ten:

10 2 = 100;
20 2 = 400;
30 2 = 900;
40 2 = 1600;
...
90 2 = 8100;
100 2 = 10 000.

We get a series of numbers:

100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10 000.

What do these numbers give us? It's simple: we get boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and greater than 40:

[Figure caption]

The same is with any other number from which you can find the square root. For example, 3364:

[Figure caption]

Thus, instead of an incomprehensible number, we get a very specific range in which the original root lies. To further narrow the scope of the search, go to the second step.

Elimination of obviously superfluous numbers

So, we have 10 numbers - candidates for the root. We received them very quickly, without complex thinking and multiplication in a column. It's time to move on.

Believe it or not, now we will reduce the number of candidate numbers to two - and again without any complicated calculations! It is enough to know the special rule. Here it is:

The last digit of the square depends only on the last digit original number.

In other words, it is enough to look at the last digit of the square - and we will immediately understand where the original number ends.

There are only 10 digits that can be in last place. Let's try to find out what they turn into when they are squared. Take a look at the table:

1	2	3	4	5	6	7	8	9	0
1	4	9	6	5	6	9	4	1	0

This table is another step towards calculating the root. As you can see, the numbers in the second line turned out to be symmetrical with respect to the five. For example:

2 2 = 4;
8 2 = 64 → 4.

As you can see, the last digit is the same in both cases. And this means that, for example, the root of 3364 necessarily ends in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

[Figure caption]

The red squares show that we don't know this figure yet. But after all, the root lies between 50 and 60, on which there are only two numbers ending in 2 and 8:

[Figure caption]

That's all! Of all the possible roots, we left only two options! And this is in the most difficult case, because the last digit can be 5 or 0. And then the only candidate for the roots will remain!

Final Calculations

So, we have 2 candidate numbers left. How do you know which one is the root? The answer is obvious: square both numbers. The one that squared will give the original number, and will be the root.

For example, for the number 3364, we found two candidate numbers: 52 and 58. Let's square them:

52 2 \u003d (50 +2) 2 \u003d 2500 + 2 50 2 + 4 \u003d 2704;
58 2 \u003d (60 - 2) 2 \u003d 3600 - 2 60 2 + 4 \u003d 3364.

That's all! It turned out that the root is 58! At the same time, in order to simplify the calculations, I used the formula of the squares of the sum and difference. Thanks to this, you didn’t even have to multiply the numbers in a column! This is another level of optimization of calculations, but, of course, it is completely optional :)

Root Calculation Examples

Theory is good, of course. But let's test it in practice.

[Figure caption]

First, let's find out between which numbers the number 576 lies:

400 < 576 < 900
20 2 < 576 < 30 2

Now let's look at the last number. It is equal to 6. When does this happen? Only if the root ends in 4 or 6. We get two numbers:

It remains to square each number and compare with the original:

24 2 = (20 + 4) 2 = 576

Excellent! The first square turned out to be equal to the original number. So this is the root.

A task. Calculate the square root:

[Figure caption]

900 < 1369 < 1600;
30 2 < 1369 < 40 2;

Let's look at the last number:

1369 → 9;
33; 37.

Let's square it:

33 2 \u003d (30 + 3) 2 \u003d 900 + 2 30 3 + 9 \u003d 1089 ≠ 1369;
37 2 \u003d (40 - 3) 2 \u003d 1600 - 2 40 3 + 9 \u003d 1369.

Here is the answer: 37.

A task. Calculate the square root:

[Figure caption]

We limit the number:

2500 < 2704 < 3600;
50 2 < 2704 < 60 2;

Let's look at the last number:

2704 → 4;
52; 58.

Let's square it:

52 2 = (50 + 2) 2 = 2500 + 2 50 2 + 4 = 2704;

We got the answer: 52. The second number will no longer need to be squared.

A task. Calculate the square root:

[Figure caption]

We limit the number:

3600 < 4225 < 4900;
60 2 < 4225 < 70 2;

Let's look at the last number:

4225 → 5;
65.

As you can see, after the second step, only one option remains: 65. This is the desired root. But let's still square it and check:

65 2 = (60 + 5) 2 = 3600 + 2 60 5 + 25 = 4225;

Everything is correct. We write down the answer.

Conclusion

Alas, no better. Let's take a look at the reasons. There are two of them:

• It is forbidden to use calculators at any normal math exam, whether it is the GIA or the Unified State Examination. And for carrying a calculator into the classroom, they can easily be kicked out of the exam.
• Don't be like stupid Americans. Which are not like roots - they cannot add two prime numbers. And at the sight of fractions, they generally get hysterical.