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Root extraction: methods, examples, solutions

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December 14, 2012

Quite often, when solving problems, we encounter large numbers from which we must extract Square root. Many students decide that this is a mistake and begin to overwrite the whole example. In no case should you do this! There are two reasons for this:

  1. The roots of large numbers are really found in tasks. 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 a powerful weapon against square roots.

  1. Limit the root to the top and bottom by multiples of 10. Thus, we will reduce the search range to 10 numbers,
  2. Of these 10 numbers, weed out those that certainly cannot be rooted. As a result, there will remain 1-2 numbers,
  3. To square these 1-2 numbers in a square. That of them, whose square is equal to the original number, will be the root.

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

Root restriction

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 the boundaries. Take, for example, the number 1296. It lies between 900 and 1600. Therefore, its root cannot be less than 30 and more than 40:

[Caption]

The same goes for any other number from which to find the square root. For example, 3364:

[Caption]

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

Elimination of obviously extra numbers

So, we have 10 numbers - candidates for the root. We got them very quickly, without complicated reflections and multiplications in a column. It's time to move on.

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

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

In other words, just look at the last digit of the square - and we will immediately understand what the original number ends with.

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

1234567890
1496569410

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:

[Caption]

The red squares indicate that we do not yet know this figure. But the root lies in the range from 50 to 60, on which there are only two numbers ending in 2 and 8:

[Caption]

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

Final calculations

So, we have 2 candidate numbers left. How to find out which one is the root? The answer is obvious: square both numbers. The one that squared gives the original number will be the root.

For example, for the number 3364, we found two candidate numbers: 52 and 58. We will square them:

52 2 = (50 +2) 2 = 2500 + 2 · 50 · 2 + 4 = 2704,
58 2 = (60 − 2) 2 = 3600 − 2 · 60 · 2 + 4 = 3364.

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

Root Calculation Examples

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

Task. Calculate the square root:

[Caption]

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

Now look at the last digit. It is 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

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

Task. Calculate the square root:

[Caption]

Hereinafter, I will write only the basic steps. So, limit the number:

We look at the last digit:

Squared:

33 2 = (30 + 3) 2 = 900 + 2 · 30 · 3 + 9 = 1089 ≠ 1369,
37 2 = (40 − 3) 2 = 1600 − 2 · 40 · 3 + 9 = 1369.

Task. Calculate the square root:

[Caption]

We look at the last digit:

Squared:

52 2 = (50 + 2) 2 = 2500 + 2 · 50 · 2 + 4 = 2704,

Received the answer: 52. It is no longer necessary to square the second number.

Task. Calculate the square root:

[Caption]

We look at the last digit:

As you can see, after the second step there is only one option left: 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,

All is correct. We write down the answer.

Conclusion

Many people ask: why take such roots at all? Isn't it better to take a calculator and not soar your brain?

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

  • It is forbidden to use calculators at any normal math exam, whether it is the State Academic Examination or the Unified State Exam. And for a calculator brought into the classroom, they can easily be kicked out of the exam.
  • Do not be like stupid Americans. Which are not like the roots - they cannot add two prime numbers. And at the sight of fractions, they generally begin to have a tantrum.

In general, learn to count. And all will be well. Good luck

What does “root extraction” mean?

We introduce the concept of root extraction.

Root extraction called finding the root value.

So, extracting the root of the nth degree from the number a is the finding of the number b whose nth degree is equal to a. When such a number b is found, it can be argued that we extracted the root.

Note that the expressions “root extraction” and “finding the root value” are equally used.

When is the root extracted?

It is said that an nth root of a retrieved for surewhen the radical number a can be represented as the nth power of some number b. For example, the cube root is extracted from the number 8, since the number 8 can be represented as the cube of the number 2. Similarly, the square root is extracted from the decimal 1.21, since 1.21 = (1,1) 2.

If the root number a is not represented in the form of the nth power of a certain number b, then they say that the root of the nth power from a is not extracted. In this case, either the written expression with the root sign does not make sense on the set of real numbers (for example, or), or the written expression makes sense, but only an approximate value of such a root can be obtained up to any decimal place. We give as an example. The square root of the number 2 is not extracted, however, its approximate value can be found accurate to any decimal place, for example, (we will consider the method of finding the values ​​of such roots in the last paragraph of this article).

Methods and examples of root extraction

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

Below we will take a look at the main methods for extracting roots.

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

If the tables are squares, cubes, etc. If it’s not at hand, then it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors. Go to the study of this method ...

We should also dwell on extracting the root from a negative number, which is possible for roots with odd exponents.

Next, we will analyze the extraction of the root from a fractional number, in particular, from an ordinary fraction, a decimal fraction and a mixed number. Go to this section ...

Finally, we consider a method that allows one to successively find the digits of the root value. To study ...

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

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

The table of squares of integers from 0 to 99 inclusive (it is shown below) consists of two zones. The first zone of the table is located on a gray background, it allows you to make a number from 0 to 99 by selecting a specific row and a specific column. For an example we will select a row of 8 tens and a column of 3 units, by this we fixed the number 83. The second zone occupies the remainder of the table. Each of its cells is at the intersection of a certain row and a certain column, and contains a square of the corresponding number from 0 to 99. At the intersection of the row of 8 tens we selected and the column of 3 units, there is a cell with the number 6,889, which is the square of the number 83.

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

Tables of squares, cubes, fourth degrees, etc. allow you to extract square roots, cubic roots, roots of the fourth degree, etc. accordingly, from the numbers in these tables. Let us explain the principle of their use in extracting roots.

Suppose we need to extract the root of the nth degree from a, and the number a is contained in the table of nth degrees. From this table we find a number b such that a = b n. Then, therefore, the number b will be the desired root of the nth power.

As an example, we show how to use the cube table to extract the cubic root of 19,683. 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 nth degrees are very convenient for extracting roots. However, they are often not at hand, and their preparation requires a certain amount of time. Moreover, it is often necessary to extract the roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.

Prime Factorization

A fairly convenient way to extract the root from a natural number (unless of course the root is extracted) is to decompose the root number into prime factors. Him the essence is as follows: after decomposing the number into prime factors, it is quite easy to represent it in the form of a degree with the necessary exponent, which allows you to get the root value. Let us explain this moment.

Suppose that a root of the nth degree is extracted from a natural number a and its value is b. In this case, the equality a = b n is true. The number b, like any natural number, can be represented as the product of all its prime factors p1, p2, ..., pm in the form p1P2· ... · pm , and the radical number a in this case is represented as (p1P2· ... · pm) n. Since the decomposition of the number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p1P2· ... · pm) n, which makes it possible to calculate the root value as.

Note that if the prime factorization of the root number a cannot be represented as (p1P2· ... · pm) n, then the root of the nth degree from this number a is not completely extracted.

We will deal with this when solving examples.

Extract the square root of 144.

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

But in the light of this paragraph, we are interested in how to extract the root by decomposing the radical number 144 into prime factors. We will analyze this method of solution.

Factor 144 into prime factors:

That is, 144 = 2 · 2 · 2 · 2 · 3 · 3. Based on the properties of a degree with a natural exponent with the obtained decomposition, one can carry out the following transformations: 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 the properties of the roots, the solution could be formalized in a slightly different way:.

.

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

Calculate the root value.

The prime factorization of the radical number 243 has the form 243 = 3 5. In this way, .

.

Is the root value an integer?

To answer this question, we decompose the radical 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 expansion is not represented as an integer cube, since the degree of a prime factor of 7 is not a multiple of three. Consequently, the cube root of the number 285,768 is not fully extracted.

Extracting roots from fractional numbers

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

Let us examine an example of extracting a root from a fraction.

What is the square root of an ordinary fraction 25/169.

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. On this, the extraction of the root from the ordinary fraction 25/169 is completed.

.

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

Extract the cubic root from decimal 474,552.

Imagine the initial decimal fraction as an ordinary fraction: 474.552 = 474552/1000. Then. It remains to extract the cubic roots located in the numerator and denominator of the obtained fraction. Since 474 552 = 2 · 2 · 2 · 3 · 3 · 3 · 13 · 13 · 13 = (2 · 3 · 13) 3 = 78 3 and 1 000 = 10 3, then. It remains only to complete the calculations.

.

Extracting a root from a negative number

Separately, it is worthwhile to dwell on the extraction of roots from negative numbers. When studying the roots, we said that when the root exponent is an odd number, then a negative number may be under the root sign. We have given the following meaning to these records: for a negative number −a and an odd root exponent 2 · n − 1, it is true. This equality gives rule for extracting roots of an odd degree from negative numbers: to extract the root from a negative number, you need to extract the root from the opposite positive number, and put a minus sign in front of the result.

Consider a solution to an example.

Find the root value.

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

Here is a brief summary of the solution: .

.

Finding the root value

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

At the first step of this algorithm, you need 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 the moment when a number is obtained that exceeds the radical number. Then the number that we raised to the power of n in the previous step will indicate the corresponding senior rank.

As an 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, 10 2 = 100> 5, which means that the category of units will be the highest digit. The value of this category, as well as the younger ones, will be found in the next steps of the root extraction algorithm.

All the following steps of the algorithm are aimed at sequentially clarifying the root value due to the fact that the values ​​of the next digits of the desired root value are found, starting from the oldest and moving to the younger ones. For example, the root value 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 digits are found.

The discharges are found by enumerating their possible values ​​0, 1, 2, ..., 9. In this case, the nth powers of the corresponding numbers are computed in parallel, and they are compared with the radical number. If at some stage the degree value exceeds the root number, then the discharge value corresponding to the previous value is considered to be found, and the transition to the next step of the root extraction algorithm is performed, if this does not happen, then the value of this discharge is 9.

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

First we find the value of the discharge units. We will sort out the values ​​0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table:

So the value of the category of units is 2 (since 2 2, and 2 3> 5). We turn to finding the value of the tenth discharge. При этом будем возводить в квадрат числа 2,0, 2,1, 2,2, …, 2,9 , сравнивая полученные значения с подкоренным числом 5 :

Так как 2,2 2 , а 2,3 2 >5 , то значение разряда десятых равно 2 . Можно переходить к нахождению значения разряда сотых:

Так найдено следующее значение корня из пяти, оно равно 2,23 . И так можно продолжать дальше находить значения : 2,236, 2,2360, 2,23606, 2,236067, … .

Для закрепления материала разберем извлечение корня с точностью до сотых при помощи рассмотренного алгоритма.

Сначала определяем старший разряд. Для этого возводим в куб числа 0, 10, 100 и т.д. until we get a number exceeding 2,151.186. We have 0 3 = 0, 10 3 = 1 000, 100 3 = 1 000 000> 2 151.186, thus, the highest digit is the discharge of tens.

Define its value.

Since 10 3, and 20 3> 2 151,186, the value of the discharge of tens is 1. We pass to units.

Thus, the value of the category of units is 2. We pass to the tenth.

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

At this stage, the root value is found with an accuracy of hundredths:.

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

Extraction of a square root using the table of squares

One of the easiest ways to calculate the roots is to using a special table. What is it and how to use it correctly?

Using the table, you can find the square of any number from 10 to 99. At the same time, tens of values ​​are in the rows of the table, and units are in the columns. The cell at the intersection of the row and column contains the square of a two-digit number. In order to calculate the square 63, you need to find a row with a value of 6 and a column with a value of 3. At the intersection, we find a cell with the number 3969.

Since root extraction is the opposite of squaring, to do this, you must do the opposite: first find the cell with the number whose radical you need to count, then determine the answer using the values ​​of the column and row. As an example, consider calculating the square root of 169.

We find the cell with this number in the table, determine dozens - 1 horizontally, and find - 3 vertically. Answer: √169 = 13.

Similarly, you can calculate the roots of cubic and nth degree using the corresponding tables.

The advantage of the method is its simplicity and the absence of additional calculations. The disadvantages are obvious: the method can only be used for a limited range of numbers (the number for which the root is located should be in the range from 100 to 9801). In addition, it will not work if the specified number is not in the table.

Factorization

If the table of squares is not at hand or it was impossible to find the root with its help, you can try factor the number under the root into prime factors. Simple factors are those that can be completely (without a remainder) divisible only by themselves or by one. Examples may be 2, 3, 5, 7, 11, 13, etc.

Consider the calculation of the root using the example of √576. We decompose it into prime factors. We get the following result: √576 = √ (2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 3 ​​∙ 3) = √ (2 ∙ 2 ∙ 2) ² ∙ √3². Using the main root property √a² = a, we get rid of the roots and squares, after which we calculate the answer: 2 ∙ 2 ∙ 2 ∙ 3 ​​= 24.

What if one of the factors does not have its own pair? For example, consider the calculation of √54. After factoring, we get the result as follows: √54 = √ (2 ∙ 3 ​​∙ 3 ∙ 3) = √3² ∙ √ (2 ∙ 3) = 3√6. The non-extractable part can be left under the root. For most problems in geometry and algebra, this answer will be counted as final. But if there is a need to calculate approximate values, you can use the methods that will be discussed later.

Heron Method

What to do when it is necessary to know at least approximately what the extracted root is equal to (if it is impossible to obtain an integer value)? The use of the Heron method gives a quick and fairly accurate result.. Its essence is to use an approximate formula:

where R is the number whose root is to be calculated, a is the nearest number whose root value is known.

Consider how the method works in practice, and evaluate how accurate it is. We calculate what √111 is equal to. The nearest number to 111 whose root is known is 121. Thus, R = 111, a = 121. We substitute the values ​​in the formula:

√111 = √121 + (111 — 121) / 2 ∙ √121 = 11 — 10 / 22 ≈ 10,55.

Now check the accuracy of the method:

The error of the method was approximately 0.3. If the accuracy of the method needs to be improved, you can repeat the steps described previously:

√111 = √111,3025 + (111 — 111,3025) / 2 ∙ √111,3025 = 10,55 — 0,3025 / 21,1 ≈ 10,536.

Check the accuracy of the calculation:

After repeated use of the formula, the error became very insignificant.

Calculation of the root by division by column

This method of finding the square root value is a little more complicated than the previous ones. However, it is the most accurate among other calculation methods without a calculator..

Suppose you need to find the square root with an accuracy of 4 decimal places. Let's analyze the calculation algorithm using an example of an arbitrary number 1308.1912.

  1. Divide the sheet of paper into 2 parts with a vertical line, and then draw another line from it to the right, slightly below the top edge. Write the number on the left side, dividing it into groups of 2 digits, moving to the right and left side of the comma. The very first digit on the left can be without a pair. If the sign is not enough on the right side of the number, then you should add 0. In our case, we get 13 08.19 12.
  2. We select the largest number whose square will be less than or equal to the first group of digits. In our case, this is 3. We write it in the upper right, 3 is the first digit of the result. We indicate 3 × 3 = 9 at the bottom right, this will be needed for subsequent calculations. Subtract 9 from 13 in the column, we get the remainder 4.
  3. We assign the next pair of numbers to the remainder of 4, we get 408.
  4. Multiply the number on the top right by 2 and write down the bottom right, adding _ x _ = to it. We get 6_ x _ =.
  5. Instead of dashes, you need to substitute the same number less than or equal to 408. We get 66 × 6 = 396. Write 6 on the top right, because this is the second digit of the result. Subtract 396 from 408 and get 12.
  6. Repeat steps 3-6. Since the numbers carried down are in the fractional part of the number, you must put the decimal point in the upper right after 6. Write the doubled result with dashes: 72_ x _ =. A suitable figure would be 1: 721 × 1 = 721. We write it in response. Subtract 1219 - 721 = 498.
  7. We follow the sequence of steps given in the previous paragraph three more times to get the required number of decimal places. If there are not enough signs for further calculations, two zeros should be added to the current number on the left.

As a result, we get the answer: √1308.1912 ≈ 36.1689. If you check the action using the calculator, you can make sure that all the signs were identified correctly.

Bitwise calculation of the square root value

The method is highly accurate.. In addition, it is understandable enough and it does not require memorizing formulas or a complex algorithm of actions, since the essence of the method is to select the correct result.

We extract the root from the number 781. We consider in detail the sequence of actions.

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