How to get a number from under the root online. How to find the square root of a number manually

Let's consider this algorithm with an example. Let's find

1st step. We divide the number under the root into two digits (from right to left):

2nd step. We extract Square root from the first face, i.e., from the number 65, we get the number 8. Under the first face, we write the square of the number 8 and subtract. We attribute the second face (59) to the remainder:

(the number 159 is the first remainder).

3rd step. We double the found root and write the result on the left:

4th step. We separate in the remainder (159) one digit on the right, on the left we get the number of tens (it is equal to 15). Then we divide 15 by the doubled first digit of the root, that is, by 16, since 15 is not divisible by 16, then in the quotient we get zero, which we write as the second digit of the root. So, in the quotient we got the number 80, which we double again, and demolish the next face

(the number 15901 is the second remainder).

5th step. We separate one digit from the right in the second remainder and divide the resulting number 1590 by 160. The result (number 9) is written as the third digit of the root and assigned to the number 160. The resulting number 1609 is multiplied by 9 and we find the following remainder (1420):

AT further action are performed in the sequence indicated in the algorithm (the root can be extracted with the required degree of accuracy).

Comment. If the root expression is a decimal fraction, then its integer part is divided into two digits from right to left, the fractional part is divided into two digits from left to right, and the root is extracted according to the specified algorithm.

DIDACTIC MATERIAL

1. Take the square root of the number: a) 32; b) 32.45; c) 249.5; d) 0.9511.

Preferably engineering - one in which there is a button with a root sign: "√". Usually, to extract the root, it is enough to type the number itself, and then press the button: “√”.

In most modern mobile phones there is a "calculator" application with a root extraction function. The procedure for finding the root of a number using a telephone calculator is similar to the above.
Example.
Find from 2.
We turn on the calculator (if it is turned off) and successively press the buttons with the image of two and the root (“2”, “√”). Pressing the "=" key is usually not necessary. As a result, we get a number like 1.4142 (the number of characters and "roundness" depends on the bit depth and calculator settings).
Note: when trying to find the root, the calculator usually gives an error.

If you have access to a computer, then finding the root of a number is very simple.
1. You can use the Calculator application available on almost any computer. For Windows XP, this program can be run as follows:
"Start" - "All Programs" - "Accessories" - "Calculator".
It is better to set the view to "normal". By the way, unlike a real calculator, the button for extracting the root is marked as "sqrt", not "√".

If you do not get to the calculator in the specified way, then you can start the standard calculator “manually”:
"Start" - "Run" - "calc".
2. To find the root of a number, you can also use some programs installed on your computer. In addition, the program has its own built-in calculator.

For example, for the MS Excel application, you can do the following sequence of actions:
We start MS Excel.

We write in any cell the number from which you want to extract the root.

Move the cell pointer to a different location

Press the function selection button (fx)

Select the "ROOT" function

As a function argument, specify a cell with a number

Press "OK" or "Enter"
The advantage of this method is that now it is enough to enter any value into the cell with a number, as in with the function immediately appears.
Note.
There are several other, more exotic ways to find the root of a number. For example, a "corner", using a slide rule or Bradis tables. However, these methods are not considered in this article due to their complexity and practical uselessness.

Related videos

Sources:

• how to find the root of a number

Sometimes there are situations when you have to perform some mathematical calculations, including extracting square roots and greater roots of a number. The "n" root of "a" is the number nth power which is the number "a".

Instruction

To find the root "n" of , do the following.

Click on your computer "Start" - "All Programs" - "Accessories". Then enter the "Utilities" subsection and select "Calculator". You can do it manually: click "Start", type "calk" in the "run" line and press "Enter". will open. To extract the square root of any number, enter this into the calculator line and press the button labeled "sqrt". The calculator will extract the root of the second degree, called the square, from the entered number.

In order to extract the root, the degree of which is higher than the second, you need to use a different kind of calculator. To do this, click the "View" button in the calculator's interface and select the "Engineering" or "Scientific" line from the menu. This type of calculator has the necessary for calculating root nth degree function.

To extract the root of the third degree (), on the "engineering" calculator, type the desired number and press the "3√" button. To obtain a root greater than 3rd, type the desired number, press the button with the icon "y√x" and then enter the number - the exponent. After that, press the equal sign ("=" button) and you will get the root you are looking for.

If your calculator does not have the "y√x" function, the following.

To extract cube root enter the root expression, then put a check mark in the check box, which is located next to the inscription "Inv". By this action, you will reverse the functions of the calculator buttons, i.e., by clicking on the button to cube, you will extract the cube root. On the button that you

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. Roots from big numbers actually occur 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 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:

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:

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:

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:

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

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

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.

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.

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, be it 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 are two prime numbers cannot fold. And at the sight of fractions, they generally get hysterical.

Sokolov Lev Vladimirovich

Objective: find and show those methods of extracting square roots that can be used without having a calculator at hand.

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Regional scientific and practical conference

students of the Tugulym city district

Extracting square roots from large numbers without a calculator

Composer: Lev Sokolov

MKOU "Tugulymskaya V (C) OSH",

8th grade

Head: Sidorova Tatiana

Nikolaevna

r.p. Tugulym, 2016

Introduction 3

Chapter 1 prime factors 4

Chapter 2

Chapter 3 two-digit numbers 6

Chapter 4

Chapter 6. Canadian Method 7

Chapter 7

Chapter 8 Odd Number Residue Method 8

Conclusion 10

References 11

Annex 12

Introduction

The relevance of research,when I studied the topic of square roots in this academic year, then I was interested in the question of how you can extract the square root of large numbers without a calculator.

I became interested and decided to study this issue deeper than it is set out in the school curriculum, and also to prepare a mini-book with the most simple ways extracting square roots from large numbers without a calculator.

Objective: find and show those methods of extracting square roots that can be used without having a calculator at hand.

Tasks:

1. Study literature on this issue.
2. Consider the features of each found method and its algorithm.
3. Show practical use acquired knowledge and evaluate

Difficulty to use various ways and algorithms.

1. Create a mini-book on the most interesting algorithms.

Object of study:mathematical symbols are square roots.

Subject of study:features of ways to extract square roots without a calculator.

Research methods:

1. Search for methods and algorithms for extracting square roots from large numbers without a calculator.
2. Comparison of the found methods.
3. Analysis of the obtained methods.

Everyone knows that taking the square root without a calculator is very difficult.

task. When there is no calculator at hand, we start using the selection method to try to remember the data from the table of squares of integers, but this does not always help. For example, the table of squares of integers does not give an answer to such questions as, for example, take the root of 75, 37,885,108,18061 and others even approximately.

Also, it is often forbidden to use a calculator at the exams of the OGE and the Unified State Examination

tables of squares of integers, but you need to take the root of 3136 or 7056, etc.

But studying the literature on this topic, I learned that to extract roots from such numbers

perhaps without a table and a calculator, people learned long before the invention of the microcalculator. Researching this topic, I found several ways to solve this problem.

Chapter 1

To extract the square root, you can decompose the number into prime factors and extract the square root from the product.

It is customary to use this method when solving tasks with roots in school.

3136│2 7056│2

1568│2 3528│2

784│2 1764│2

392│2 882│2

196│2 441│3

98│2 147│3

49│7 49│7

7│7 7│7

√3136 = √2²∙2²∙2²∙7² = 2∙2∙2∙7 = 56 √3136 = √2²∙2²∙3²∙7² = 2∙2∙3∙7 = 84

Many use it successfully and consider it the only one. Extracting a root by factoring is a laborious task, which also does not always lead to the desired result. Try to extract the square root of the number 209764? Decomposition into prime factors gives the product 2∙2∙52441. And how to be further? Everyone faces this problem, and calmly write down the remainder of the expansion under the root sign in the answer. By trial and error, by selection, decomposition, of course, can be done if you are sure that you will get a beautiful answer, but practice shows that tasks with complete decomposition are very rarely offered. More often we see that the root cannot be fully extracted.

Therefore, this method only partially solves the problem of extracting without a calculator.

Chapter 2

To extract the square root with a corner andLet's look at the algorithm:
1st step. The number 8649 is divided into faces from right to left; each of which must contain two digits. We get two edges:.
2nd step. We extract the square root of the first face 86, we getwith a disadvantage. The number 9 is the first digit of the root.
3rd step. The number 9 is squared (9 2 = 81) and the number 81 is subtracted from the first face, we get 86- 81=5. The number 5 is the first remainder.
4th step. To the remainder 5 we attribute the second face 49, we get the number 549.

5th step . We double the first digit of the root of 9 and, writing on the left, we get -18

It is necessary to attribute such a largest digit to the number so that the product of the number that we get by this digit would either be equal to the number 549 or less than 549. This is the number 3. It is found by selection: the number of tens of the number 549, that is, the number 54 is divided by 18, we get 3, since 183 ∙ 3 \u003d 549. The number 3 is the second digit of the root.

6th step. We find the remainder 549 - 549 = 0. Since the remainder is zero, we got the exact value of the root - 93.

I will give another example: extract √212521

Chapter 3

I learned about this method from the Internet. The method is very simple and gives instant extraction of the square root of any integer from 1 to 100 with an accuracy of tenths without a calculator. One condition for this method is the presence of a table of squares of numbers up to 99.

(It is in all grade 8 algebra textbooks, and is offered as reference material on the OGE exam.)

Open the table and check the speed of finding the answer. But first, a few recommendations: the leftmost column - these will be integers in the answer, the topmost line - these are the tenths in the answer. And then everything is simple: close the last two digits of the number in the table and find the number you need, not exceeding the root number, and then follow the rules of this table.

Let's look at an example. Let's find the value √87.

We close the last two digits for all numbers in the table and find close ones for 87 - there are only two of them 86 49 and 88 37. But 88 is already a lot.

So, there is only one thing left - 8649.

The left column gives the answer 9 (these are integers), and the top line is 3 (these are tenths). So √87≈ 9.3. Let's check on MK √87 ≈ 9.327379.

Quick, easy, affordable on the exam. But it is immediately clear that roots greater than 100 cannot be extracted by this method. The method is convenient for tasks with small roots and in the presence of a table.

Chapter 4

The ancient Babylonians used the following method to find the approximate value of the square root of their x number. They represented the number x as the sum of a 2 + b, where a 2 the exact square of a natural number a (a 2 . (1)

Using formula (1), we extract the square root, for example, from the number 28:

The result of extracting the root of 28 using MK 5.2915026.

As we can see, the way of the Babylonians gives a good approximation to exact value root.

Chapter 5

(only for four digit numbers)

It’s worth clarifying right away that this method is applicable only for extracting the square root from an exact square, and the finding algorithm depends on the value of the root number.

1. Extracting roots up to the number 75 2 = 5625

For example: √¯3844 = √¯ 37 00 + 144 = 37 + 25 = 62.

We represent the number 3844 as a sum by selecting the square 144 from this number, then we discard the selected square, tothe number of hundreds of the first term(37) always add 25 . We get the answer 62.

So you can only take square roots up to the number 75 2 =5625!

2) Extracting roots after the number 75 2 = 5625

How to verbally extract square roots from numbers greater than 75 2 =5625?

For example: √7225 = √ 70 00 + 225 = 70 + √225 = 70 + 15 = 85.

To clarify, 7225 is represented as the sum of 7000 and the highlighted square 225. Thenadd the square root to the hundreds out of 225, equal to 15.

We get the answer 85.

This way of finding is very interesting and to some extent original, but in the course of my research I met only once in the work of a Perm teacher.

Perhaps it is little studied or has some exceptions.

It is quite difficult to remember due to the duality of the algorithm and is applicable only for four-digit numbers of exact roots, but I have worked through many examples and made sure that it is correct. In addition, this method is available to those who have already memorized the squares of numbers from 11 to 29, because without their knowledge it will be useless.

Chapter 6

√ X = √ S + (X - S) / (2 √ S) where X is the number to take the square root of and S is the number of the nearest perfect square.

Let's try to take the square root of 75

√ 75 = 9 + (- 6/18) = 9 - 0,333 = 8,667

With a detailed study of this method, one can easily prove its similarity with the Babylonian and argue for the copyright of the invention of this formula, if any, in reality. The method is simple and convenient.

Chapter 7

This method is offered English students Mathematical College London, but everyone in his life at least once involuntarily used this method. It is based on selection different values squares of close numbers by narrowing the search area. Everyone can master this method, but it’s unlikely to use it, because it requires repeated calculation of the product of a column of not always correctly guessed numbers. This method loses both in the beauty of the solution and in time. The algorithm is simple:

Let's say you want to take the square root of 75.

Since 8 2 = 64 and 9 2 = 81, you know, the answer is somewhere in between.

Try to erect 8.5 2 and you get 72.25 (too little)

Now try 8.6 2 and you get 73.96 (too small, but getting closer)

Now try 8.7 2 and you get 75.69 (too big)

Now you know the answer is between 8.6 and 8.7

Try to erect 8.65 2 and you get 74.8225 (too little)

Now try 8.66 2 ... and so on.

Keep going until you get an answer that's accurate enough for you.

Chapter 8 Odd Number Subtraction Method

Many people know the method of extracting the square root by decomposing a number into prime factors. In my work, I will present another way by which you can find out the integer part of the square root of a number. The method is very simple. Note that the following equalities are true for the squares of numbers:

1=1 2

1+3=2 2

1+3+5=3 2

1+3+5+7=4 2 etc.

Rule: you can find out the integer part of the square root of a number by subtracting from it all odd numbers in order, until the remainder is less than the next subtracted number or equal to zero, and counting the number of actions performed.

For example, to get the square root of 36 and 121 is:

Total number of subtractions = 6, so the square root of 36 = 6.

Total subtractions = 11, so √121 = 11.

Another example: find √529

Solution: 1)_529

2)_528

3)_525

4)_520

5)_513

6)_504

7)_493

8)_480

9)_465

10)_448

11)_429

12)_408

13)_385

14)_360

15)_333

16)_304

17)_273

18)_240

19)_205

20)_168

21)_129

22)_88

23)_45

Answer: √529 = 23

Scientists call this method the arithmetic extraction of the square root, and behind the eyes "the turtle method" because of its slowness.
The disadvantage of this method is that if the extracted root is not an integer, then you can find out only its integer part, but not more accurately. At the same time, this method is quite accessible to children who solve the simplest problems. math problems, requiring the extraction of the square root. Try extracting the square root of a number like 5963364 in this way and you will find that it "works", certainly without errors for exact roots, but very, very long in solution.

Conclusion

The root extraction methods described in the paper are found in many sources. However, sorting them out turned out to be for me daunting task, which aroused considerable interest. The presented algorithms will allow everyone who is interested in this topic to quickly master the skills of calculating the square root, they can be used to check your solution and not depend on a calculator.

As a result of the research, I came to the conclusion: various methods of extracting the square root without a calculator are necessary in the school mathematics course in order to develop calculation skills.

The theoretical significance of the study - the main methods for extracting square roots are systematized.

Practical significance:in creating a mini-book containing a reference scheme for extracting square roots in various ways (Appendix 1).

Literature and Internet sites:

1. I.N. Sergeev, S.N. Olechnik, S.B. Gashkov "Apply Mathematics". - M.: Nauka, 1990
2. Kerimov Z., "How to find a whole root?" Popular science physics and mathematics journal "Kvant" №2, 1980
3. Petrakov I.S. "math circles in grades 8-10"; The book for the teacher.

–M.: Enlightenment, 1987

1. Tikhonov A.N., Kostomarov D.P. "Stories about applied mathematics" - M.: Nauka. Main edition of physical and mathematical literature, 1979
2. Tkacheva M.V. Home mathematics. Book for 8th grade students educational institutions. - Moscow, Enlightenment, 1994.
3. Zhokhov V.I., Pogodin V.N. Reference tables in mathematics. - M .: LLC "Publishing house" ROSMEN-PRESS ", 2004.-120 p.
4. http://translate.google.ru/translate
5. http://www.murderousmaths.co.uk/books/sqroot.htm
6. http://en.wikipedia.ord/wiki/theorema/

Good afternoon, dear guests!

My name is Lev Sokolov, I'm in the 8th grade at an evening school.

I present to your attention the work on the topic:Extracting square roots from large numbers without a calculator.

When studying a topicsquare roots this academic year, I was interested in the question of how you can extract the square root of large numbers without a calculator and I decided to study it deeper, because on next year I have to take a math exam.

The purpose of my work:find and show ways to extract square roots without a calculator

To achieve the goal, I solved the following tasks:

1. Study the literature on this issue.

2. Consider the features of each found method and its algorithm.

3. Show the practical application of the acquired knowledge and assess the degree of difficulty in using various methods and algorithms.

4.Create a mini book according to the most interesting algorithms.

The object of my research wassquare roots.

Subject of study:ways to extract square roots without a calculator.

Research methods:

1. Search for methods and algorithms for extracting square roots from large numbers without a calculator.

2. Comparison and analysis of the methods found.

I found and studied 8 ways to extract square roots without a calculator and put them into practice. The names of the methods found are given on the slide.

I will focus on those that I liked.

I will show by example how it is possible to extract the square root of the number 3025 by means of decomposition into prime factors.

The main disadvantage of this method- it takes a lot of time.

Using the formula of Ancient Babylon, I will extract the square root of the same number 3025.

The method is convenient only for small numbers.

From the same number 3025 we extract the square root with a corner.

In my opinion, this is the most universal way, it applies to any numbers.

AT modern science there are many ways to extract the square root without a calculator, but I have not studied everything.

The practical significance of my work:in the creation of a mini-book containing a reference scheme for extracting square roots in various ways.

The results of my work can be successfully applied in the lessons of mathematics, physics and other subjects where extraction of roots is required without a calculator.

Thank you for your attention!

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Extracting square roots from large numbers without a calculator Performer: Lev Sokolov, MKOU "Tugulymskaya V (C) OSH", 8th grade Supervisor: Sidorova Tatyana Nikolaevna I category, teacher of mathematics r.p. Tugulym

The correct application of methods can be learned by applying and using a variety of examples. G. Zeiten The purpose of the work: to find and show those methods of extracting square roots that can be used without having a calculator at hand. Tasks: - To study the literature on this issue. - Consider the features of each found method and its algorithm. - Show the practical application of the acquired knowledge and assess the degree of difficulty in using various methods and algorithms. - Create a mini-book on the most interesting algorithms.

Object of study: square roots Subject of study: methods of extracting square roots without a calculator. Research methods: Search for methods and algorithms for extracting square roots from large numbers without a calculator. Comparison of the found methods. Analysis of the obtained methods.

Square root methods: 1. Prime factorization method 2. Corner square root extraction 3. Two-digit square root method 4. Ancient Babylon formula 5. Full square rejection method 6. Canadian method 7. Guessing method 8. Reduction method odd number

Prime factorization method To extract the square root, you can factorize a number into prime factors and extract the square root of the product. 3136 │2 7056 │2 209764 │2 1568 │2 3528 │2 104882 │2 784 │2 1764│2 52441 │229 392 │2 882 │229 196 │2 441 │ │2 2 147 √209764 = √2∙2∙52441 = 49│7 49│7 = √2²∙229² = 458 √7056 = √2²∙2²∙3²∙7² = 2∙2∙3∙7 = 84. It is not always easy to decompose, more often it is not completely removed, it takes a lot of time.

Formula of Ancient Babylon (Babylonian method) An algorithm for extracting the square root using the ancient Babylonian method. one . Represent the number c as a sum a ² + b, where a ² is the closest to the number c the exact square of the natural number a (a ² ≈ c); 2. The approximate value of the root is calculated by the formula: The result of extracting the root using the calculator is 5.292.

Extracting the square root with a corner The method is almost universal, since it is applicable to any numbers, but compiling a rebus (guessing the number at the end of the number) requires logic and good computing skills in a column.

Algorithm for extracting the square root with a corner 1. Divide the number (5963364) into pairs from right to left (5`96`33`64) 2. Extract the square root from the first left group (- number 2). So we get the first digit of the number. 3. Find the square of the first digit (2 2 \u003d 4). 4. Find the difference between the first group and the square of the first digit (5-4=1). 5. We demolish the next two digits (we got the number 196). 6. We double the first figure we found, write it down to the left behind the line (2*2=4). 7. Now you need to find the second digit of the number: the doubled first digit that we found becomes the digit of the tens of the number, when multiplied by the number of units, you need to get a number less than 196 (this is the number 4, 44 * 4 \u003d 176). 4 is the second digit of &. 8. Find the difference (196-176=20). 9. We demolish the next group (we get the number 2033). 10. We double the number 24, we get 48. 11. 48 tens in the number, when multiplied by the number of units, we should get a number less than 2033 (484 * 4 \u003d 1936). The number of units found by us (4) is the third digit of the number. Then the process is repeated.

Odd number subtraction method (arithmetic method) Square root algorithm: Subtract odd numbers in order until the remainder is less than the next number to be subtracted or equal to zero. Count the number of actions performed - this number is the integer part of the number of the extracted square root. Example 1: Calculate 1. 9 − 1 = 8; 8 − 3 = 5; 5 − 5 = 0. 2. 3 steps completed

36 - 1 = 35 - 3 = 32 - 5 = 27 - 7 = 20 - 9 = 11 - 11 = 0 total subtractions = 6, so the square root of 36 = 6. 121 - 1 = 120 - 3 = 117 - 5 = 112 - 7 = 105 - 9 = 96 - 11 = 85 - 13 = 72 - 15 = 57 - 17 = 40 - 19 = 21 - 21 = 0 Total number of subtractions = 11, so the square root of 121 = 11. 5963364 = ??? Russian scientists "behind their backs" call it the "tortoise method" because of its slowness. It is inconvenient for large numbers.

The theoretical significance of the study - the main methods for extracting square roots are systematized. Practical significance: in the creation of a mini-book containing a reference scheme for extracting square roots in various ways.

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When solving some problems, you will need to take the square root of a large number. How to do it?

Odd number subtraction method.

The method is very simple. Note that the following equalities are true for the squares of numbers:

1=1 2

1+3=2 2

1+3+5=3 2

1+3+5+7=4 2 etc.

Rule: you can find out the integer part of the square root of a number by subtracting from it all odd numbers in order, until the remainder is less than the next subtracted number or equal to zero, and counting the number of actions performed.

For example, to get the square root of 36 and 121 is:

36 - 1 = 35 - 3 = 32 - 5 = 27 - 7 = 20 - 9 = 11 - 11 = 0

Total number of subtractions = 6, so the square root of 36 = 6.

121 - 1 = 120 - 3 = 117- 5 = 112 - 7 = 105 - 9 = 96 - 11 = 85 – 13 = 72 - 15 = 57 – 17 = 40 - 19 = 21 - 21 = 0

Total number of subtractions = 11, so√121 = 11.

Canadian method.

This fast method was opened by young scientists from one of the leading universities in Canada in the 20th century. Its accuracy is no more than two or three decimal places. Here is their formula:

√ X = √ S + (X - S) / (2 √ S), where X is the number to square the root of, and S is the number of the nearest perfect square.

Example. Take the square root of 75.

X = 75, S = 81. This means that √ S = 9.

Let's calculate √75 using this formula: √ 75 = 9 + (75 - 81) / (2∙ 9)
√ 75 = 9 + (- 6/18) = 9 - 0,333 = 8,667

A method for extracting the square root with a corner.

1. Split the number (5963364) into pairs from right to left (5`96`33`64)

2. We extract the square root of the first group on the left (- number 2). So we get the first digit of the number.

3. Find the square of the first digit (2 2 =4).

4. Find the difference between the first group and the square of the first digit (5-4=1).

5. We demolish the next two digits (we got the number 196).

6. We double the first figure we found, write it down to the left behind the line (2*2=4).

7. Now you need to find the second digit of the number: the doubled first digit that we found becomes the digit of the tens of the number, when multiplied by the number of units, you need to get a number less than 196 (this is the number 4, 44 * 4 \u003d 176). 4 is the second digit of &.

8. Find the difference (196-176=20).

9. We demolish the next group (we get the number 2033).

10. Double the number 24, we get 48.

11.48 tens in a number, when multiplied by the number of units, we should get a number less than 2033 (484 * 4 \u003d 1936). The number of units found by us (4) is the third digit of the number.

Action square root extractionthe opposite of squaring.

√81= 9 9 2 =81.

selection method.

Example: Extract the root of the number 676.

We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20

Exact squares natural numbers end with 0; one; 4; 5; 6; nine.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.

Left to check: 24 2 = 576, 26 2 = 676.

Answer: √ 676 = 26.

Another example: √6889 .

Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80 The number 9 is given by 3 2 and 7 2 , then √6889 is either 83 or 87.

Check: 83 2 = 6889.

Answer: √6889 = 83.

If you find it difficult to solve by the selection method, then you can factorize the root expression.

For example, find √893025 .

Let's factorize the number 893025, remember, you did it in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

Babylonian method.

Step #1. Express the number x as a sum: x=a 2 + b, where a 2 the nearest exact square of a natural number a to x.

Step #2. Use formula:

Example. Calculate .

arithmetic method.

We subtract from the number all odd numbers in order, until the remainder is less than the next number to be subtracted or equal to zero. Having counted the number of actions performed, we determine the integer part of the square root of the number.

Example. Calculate the integer part of a number.

Decision. 12 - 1 = 11; 11 - 3 = 8; 8 - 5 = 3; 3 3 - integer part of the number. So, .

Method (known as Newton's method)is as follows.

Let a 1 - first approximation of a number(as a 1 you can take the values ​​of the square root of a natural number - an exact square that does not exceed .

This method allows you to extract the square root of a large number with any accuracy, though with a significant drawback: the cumbersomeness of calculations.

Assessment method.

Step #1. Find out the range in which the original root lies (100; 400; 900; 1600; 2500; 3600; 4900; 6400; 8100; 10,000).

Step #2. By the last digit, determine which digit the desired number ends with.

Step #3. Square the expected numbers and determine the desired number from them.

Example 1. Calculate .

Decision. 2500 50 2 2 50

= *2 or = *8.

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

Therefore, = 58.

Students always ask: “Why can't I use a calculator on a math exam? How to extract the square root of a number without a calculator? Let's try to answer this question.

How to extract the square root of a number without the help of a calculator?

Action square root extraction the opposite of squaring.

√81= 9 9 2 =81

If from positive number take the square root and square the result, we get the same number.

From small numbers that are exact squares of natural numbers, for example 1, 4, 9, 16, 25, ..., 100, square roots can be extracted verbally. Usually at school they teach a table of squares of natural numbers up to twenty. Knowing this table, it is easy to extract the square roots from the numbers 121,144, 169, 196, 225, 256, 289, 324, 361, 400. From numbers greater than 400, you can extract using the selection method using some tips. Let's try an example to consider this method.

Example: Extract the root of the number 676.

We notice that 20 2 \u003d 400, and 30 2 \u003d 900, which means 20< √676 < 900.

Exact squares of natural numbers end in 0; one; 4; 5; 6; nine.
The number 6 is given by 4 2 and 6 2 .
So, if the root is taken from 676, then it is either 24 or 26.

It remains to check: 24 2 = 576, 26 2 = 676.

Answer: √676 = 26 .

More example: √6889 .

Since 80 2 \u003d 6400, and 90 2 \u003d 8100, then 80< √6889 < 90.
The number 9 is given by 3 2 and 7 2, then √6889 is either 83 or 87.

Check: 83 2 = 6889.

Answer: √6889 = 83 .

If you find it difficult to solve by the selection method, then you can factorize the root expression.

For example, find √893025.

Let's factorize the number 893025, remember, you did it in the sixth grade.

We get: √893025 = √3 6 ∙5 2 ∙7 2 = 3 3 ∙5 ∙7 = 945.

More example: √20736. Let's factorize the number 20736:

We get √20736 = √2 8 ∙3 4 = 2 4 ∙3 2 = 144.

Of course, factoring requires knowledge of divisibility criteria and factoring skills.

And finally, there is square root rule. Let's look at this rule with an example.

Calculate √279841.

To extract the root of a multi-digit integer, we split it from right to left into faces containing 2 digits each (there may be one digit in the left extreme face). Write like this 27'98'41

To get the first digit of the root (5), we extract the square root of the largest exact square contained in the first left face (27).
Then the square of the first digit of the root (25) is subtracted from the first face and the next face (98) is attributed (demolished) to the difference.
To the left of the resulting number 298, they write the double digit of the root (10), divide by it the number of all tens of the previously obtained number (29/2 ≈ 2), experience the quotient (102 ∙ 2 = 204 should not be more than 298) and write (2) after the first digit of the root.
Then the resulting quotient 204 is subtracted from 298, and the next facet (41) is attributed (demolished) to the difference (94).
To the left of the resulting number 9441, they write the double product of the digits of the root (52 ∙ 2 = 104), divide by this product the number of all tens of the number 9441 (944/104 ≈ 9), experience the quotient (1049 ∙ 9 = 9441) should be 9441 and write it down (9) after the second digit of the root.

We got the answer √279841 = 529.

Similarly extract roots of decimals. Only the radical number must be divided into faces so that the comma is between the faces.

Example. Find the value √0.00956484.

You just have to remember that if decimal has an odd number of decimal places, it does not take exactly the square root.

So, now you have seen three ways to extract the root. Choose the one that suits you best and practice. To learn how to solve problems, you need to solve them. And if you have any questions, sign up for my lessons.

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