How to find the root of five. How to extract the root of a multi-digit number

Do you want to do well on the Unified State Examination in mathematics? Then you need to be able to count quickly, correctly and without a calculator. After all main reason loss of points on the Unified State Exam in mathematics - computational errors.

According to the rules conducting the Unified State Exam, it is prohibited to use a calculator during the mathematics exam. The price may be too high - removal from the exam.

In fact, you don’t need a calculator for the Unified State Examination in mathematics. All problems are solved without it. The main thing is attention, accuracy and some secret techniques, which we will tell you about.

Let's start with the main rule. If a calculation can be simplified, simplify it.

Here, for example, is the “devilish equation”:

Seventy percent of graduates solve it head-on. They calculate the discriminant using the formula, after which they say that the root cannot be extracted without a calculator. But you can divide the left and right sides of the equation by . It will work out

Which way is easier? :-)

Many schoolchildren do not like columnar multiplication. Nobody liked solving boring “examples” in fourth grade. However, in many cases it is possible to multiply numbers without a “column”, in a row. It's much faster.

Please note that we do not start with smaller digits, but with larger ones. It's comfortable.

Now - division. It is not easy to divide “in a column” by . But remember that the division sign: and the fractional bar are the same thing. Let's write it as a fraction and reduce the fraction:

Another example.

How to square quickly and without any columns two-digit number? We apply abbreviated multiplication formulas:

Sometimes it is convenient to use another formula:

Numbers ending in , are squared instantly.

Let's say we need to find the square of a number ( - not necessarily a number, any natural number). We multiply by and add to the result. All!

For example: (and attributed).

(and attributed).

(and attributed).

This method is useful not only for squaring, but for taking the square root of numbers ending in .

How do you even extract it? Square root without a calculator? We'll show you two ways.

The first method is to factorize the radical expression.

For example, let's find
A number is divisible by (since the sum of its digits is divisible by ). Let's factorize:

Let's find it. This number is divisible by . It is also divided by. Let's factor it out.

Another example.

There is a second way. It is convenient if the number from which you need to extract the root cannot be factorized.

For example, you need to find . The number under the root is odd, it is not divisible by, is not divisible by, is not divisible by... You can continue to look for what it is divisible by, or you can do it easier - find this root by selection.

Obviously, a two-digit number was squared, which is between the numbers and , since , , and the number is between them. We already know the first digit in the answer, it is .

The last digit in the number is . Because the , , last digit the answer is either , or . Let's check:
. Happened!

Let's find it.

This means that the first digit in the answer is five.

The last digit in the number is nine. , . This means that the last digit in the answer is either , or .

Let's check:

If the number from which you need to extract the square root ends in or, then the square root of it will be an irrational number. Because no integer square ends in or . Remember that in the tasks part Unified State Exam options in mathematics, the answer must be written as an integer or finite number decimal, that is, must be a rational number.

We encounter quadratic equations in problems and variants of the Unified State Examination, as well as in parts. They need to count the discriminant and then extract the root from it. And it is not at all necessary to look for roots from five digit numbers. In many cases, the discriminant can be factorized.

For example, in Eq.

Another situation in which the expression under the root can be factorized is taken from the problem.

Hypotenuse right triangle is equal to , one of the legs is equal to , find the second leg.

According to the Pythagorean theorem, it is equal to . You can count in a column for a long time, but it’s easier to use the abbreviated multiplication formula.

And now we’ll tell you the most interesting thing - why graduates lose precious points on the Unified State Exam. After all, errors in calculations do not just happen.

1 . Right way To lose points - sloppy calculations, in which something is corrected, crossed out, one number is written on top of another. Look at your drafts. Perhaps they look the same? :-)

Write legibly! Don't skimp on paper. If something is wrong, do not correct one number for another, it is better to write it again.

2. For some reason, many schoolchildren, when counting in a column, try to do it 1) very, very quickly, 2) in very small numbers, in the corner of their notebook, and 3) with a pencil. The result is this:

It's impossible to make anything out. So is it any surprise that the Unified State Exam score is lower than expected?

3. Many schoolchildren are accustomed to ignoring parentheses in expressions. Sometimes this happens:

Remember that the equal sign is not placed just anywhere, but only between equal amounts. Write correctly, even in draft form.

4 . Great amount computational errors associated with fractions. If you are dividing a fraction by a fraction, use what
A “hamburger” is drawn here, that is multi-story fraction. It is extremely difficult to get the correct answer using this method.

Let's summarize.

Checking the tasks of the first part profile Unified State Examination in mathematics - automatic. There is no “almost right” answer here. Either he is correct or he is not. One computational error - and hello, the task does not count. Therefore, it is in your interests to learn to count quickly, correctly and without a calculator.

The tasks of the second part of the profile Unified State Examination in mathematics are checked by an expert. Take care of him! Let him understand both your handwriting and the logic of the decision.

The circle showed how you can extract square roots in a column. You can calculate the root with arbitrary precision, find any number of digits in it decimal notation, even if it turns out to be irrational. The algorithm was remembered, but questions remained. It was not clear where the method came from and why it gave the correct result. It wasn’t in the books, or maybe I was just looking in the wrong books. In the end, like much of what I know and can do today, I came up with it myself. I share my knowledge here. By the way, I still don’t know where the rationale for the algorithm is given)))

So, first I tell you “how the system works” with an example, and then I explain why it actually works.

Let’s take a number (the number was taken “out of thin air”, it just came to mind).

1. We divide its numbers into pairs: those to the left of decimal point, we group two from right to left, and those to the right - two from left to right. We get.

2. We extract the square root from the first group of numbers on the left - in our case this is (it is clear that the exact root may not be extracted, we take a number whose square is as close as possible to our number formed by the first group of numbers, but does not exceed it). In our case this will be a number. We write down the answer - this is the most significant digit of the root.

3. We square the number that is already in the answer - this - and subtract it from the first group of numbers on the left - from the number. In our case it remains .

4. We assign the following group of two numbers to the right: . We multiply the number that is already in the answer by , and we get .

5. Now watch carefully. We need to assign one digit to the number on the right, and multiply the number by, that is, by the same assigned digit. The result should be as close as possible to, but again no more than this number. In our case, this will be the number, we write it in the answer next to, on the right. This is the next digit in the decimal notation of our square root.

6. From subtract the product , we get .

7. Next, we repeat the familiar operations: we assign the following group of digits to the right, multiply by , to the resulting number > we assign one digit to the right, such that when multiplied by it we get a number smaller than , but closest to it - this is the next digit in decimal root notation.

The calculations will be written as follows:

And now the promised explanation. The algorithm is based on the formula

Comments: 50

1. 2 Anton:

   Too chaotic and confusing. Arrange everything point by point and number them. Plus: explain where we substitute in each action required values. I’ve never calculated a root root before; I had a hard time figuring it out.

2. 5 Julia:

3. 6 :

   Yulia, 23 on this moment written on the right, these are the first two (on the left) already obtained digits of the root in the answer. Multiply by 2 according to the algorithm. We repeat the steps described in point 4.

4. 7 zzz:

   error in “6. From 167 we subtract the product 43 * 3 = 123 (129 nada), we get 38.”
   I don’t understand how it turned out to be 08 after the decimal point...

5. 9 Fedotov Alexander:

   And even in the pre-calculator era, we were taught at school not only square, but also cube root extract in a column, but this is more tedious and painstaking work. It would be easier to use Bradis tables or slide rule, which we already studied in high school.

6. 10 :

   Alexander, you are right, you can extract it into a column and roots higher degrees. I'm going to write just about how to find the cube root.

7. 12 Sergei Valentinovich:

   Dear Elizaveta Alexandrovna! In the late 70s, I developed a scheme for automatic (i.e., not by selection) calculation of quadra. root on the Felix adding machine. If you are interested, I can send you a description.

8. 14 Vlad aus Engelsstadt:

   (((Extracting the square root of the column)))
   The algorithm is simplified if you use the 2nd number system, which is studied in computer science, but is also useful in mathematics. A.N. Kolmogorov in popular lectures I gave this algorithm for schoolchildren. His article can be found in the “Chebyshev Collection” (Mathematical Journal, look for a link to it on the Internet)
   By the way, say:
   G. Leibniz at one time toyed with the idea of ​​​​transitioning from the 10th number system to the binary one because of its simplicity and accessibility for beginners ( junior schoolchildren). But breaking established traditions is like breaking a fortress gate with your forehead: it’s possible, but it’s useless. So it turns out that according to the most cited in old times to the bearded philosopher: the traditions of all dead generations suppress the consciousness of the living.

   Until next time.

9. 15 Vlad aus Engelsstadt:

   ))Sergey Valentinovich, yes, I’m interested...((

   I bet that this is a variation on the “Felix” of the Babylonian method of extracting the square knight using the method of successive approximations. This algorithm was covered by Newton's method (tangent method)

   I wonder if I was wrong in my forecast?

10. 18 :

    2Vlad aus Engelsstadt

    Yes, the algorithm in binary should be simpler, that's pretty obvious.

    About Newton's method. Maybe that's true, but it's still interesting

11. 20 Kirill:

    Thanks a lot. But there is still no algorithm, no one knows where it came from, but the result is correct. THANKS A LOT! I've been looking for this for a long time)

12. 21 Alexander:

    How will you extract the root from a number where the second group from left to right is very small? for example, everyone's favorite number is 4,398,046,511,104. After the first subtraction, it is not possible to continue everything according to the algorithm. Can you explain please.

13. 22 Alexey:

    Yes, I know this method. I remember reading it in the book “Algebra” of some old edition. Then, by analogy, he himself deduced how to extract the cube root in a column. But there it’s already more complicated: each digit is determined not by one (as for a square), but by two subtractions, and even there you have to multiply long numbers every time.

14. 23 Artem:

    There are typos in the example of extracting the square root of 56789.321. The group of numbers 32 is assigned twice to the numbers 145 and 243, in the number 2388025 the second 8 must be replaced by 3. Then the last subtraction should be written as follows: 2431000 – 2383025 = 47975.
    Additionally, when dividing the remainder by the doubled value of the answer (ignoring the comma), we get the additional quantity significant figures(47975/(2*238305) = 0.100658819...), which should be added to the answer (√56789.321 = 238.305... = 238.305100659).

15. 24 Sergey:

    Apparently the algorithm came from Isaac Newton’s book “General Arithmetic or a book on arithmetic synthesis and analysis.” Here is an excerpt from it:

    ABOUT EXTRACTING ROOTS

    To extract the square root of a number, first of all you should place a dot over its digits, starting with units. Then you should write in the quotient or radical the number whose square is equal to or closest in disadvantage to the numbers or number preceding the first point. After subtracting this square, the remaining digits of the root will be sequentially found by dividing the remainder by twice the value of the already extracted part of the root and subtracting each time from the remainder of the square the last found digit and its tenfold product by the named divisor.

16. 25 Sergey:

    Please also correct the title of the book “General Arithmetic or a book about arithmetic synthesis and analysis”

17. 26 Alexander:

    thanks for interesting material. But this method seems to me somewhat more complicated than what is needed, for example, for a schoolchild. I use a simpler method based on decomposition quadratic function using the first two derivatives. Its formula is:
    sqrt(x)= A1+A2-A3, where
    A1 is the integer whose square is closest to x;
    A2 is a fraction, the numerator is x-A1, the denominator is 2*A1.
    For most numbers found in school course, this is enough to get the result accurate to the hundredth.
    If you need a more accurate result, take
    A3 is a fraction, the numerator is A2 squared, the denominator is 2*A1+1.
    Of course, to use it you need a table of squares of integers, but this is not a problem at school. Remembering this formula is quite simple.
    However, it confuses me that I obtained A3 experimentally as a result of experiments with spreadsheet and I don’t quite understand why this member looks like this. Maybe you can give me some advice?

18. 27 Alexander:

    Yes, I've considered these considerations too, but the devil is in the details. You write:
    “since a2 and b differ quite little.” The question is exactly how little.
    This formula works well on numbers in the second ten and much worse (not up to hundredths, only up to tenths) on numbers in the first ten. Why this happens is difficult to understand without the use of derivatives.

19. 28 Alexander:

    I will clarify what I see as the advantage of the formula I propose. It does not require the not entirely natural division of numbers into pairs of digits, which, as experience shows, is often performed with errors. Its meaning is obvious, but for a person familiar with analysis, it is trivial. Works well on numbers from 100 to 1000, which are the most common numbers encountered in school.

20. 29 Alexander:

    By the way, I did some digging and found the exact expression for A3 in my formula:
    A3= A22 /2(A1+A2)

21. 30 vasil stryzhak:

    Nowadays, widespread use computer technology, the question of extracting a square knight from a number is not worth it from a practical point of view. But for mathematics lovers, various options for solving this problem will undoubtedly be of interest. IN school curriculum way of this calculation without involvement additional funds should take place on a par with multiplication and long division. The calculation algorithm must not only be memorized, but also understandable. Classic method, provided in this material for discussion with disclosure of the essence, fully complies with the above criteria.
    A significant drawback of the method proposed by Alexander is the use of a table of squares of integers. The author is silent about the majority of numbers encountered in the school course. As for the formula, in general I like it due to the relatively high accuracy of the calculation.

22. 31 Alexander:

    for 30 vasil stryzhak
    I didn't keep anything quiet. The table of squares is supposed to be up to 1000. In my time at school they simply learned it by heart and it was in all mathematics textbooks. I explicitly named this interval.
    As for computer technology, it is not used mainly in mathematics lessons, unless the topic of using a calculator is specifically discussed. Calculators are now built into devices that are prohibited for use on the Unified State Exam.

23. 32 vasil stryzhak:

    Alexander, thanks for the clarification! I thought that for the proposed method it is theoretically necessary to remember or use a table of squares of all two-digit numbers. Then for radical numbers not included in the interval from 100 to 10000, you can use the technique of increasing or decreasing them by required amount orders of comma transfer.

24. 33 vasil stryzhak:

25. 39 ALEXANDER:

    MY FIRST PROGRAM IN IAMB LANGUAGE ON THE SOVIET MACHINE “ISKRA 555″ WAS WRITTEN TO EXTRACT THE SQUARE ROOT OF A NUMBER USING THE COLUMN EXTRACTION ALGORITHM! and now I forgot how to extract it manually!

Extracting the root from large number. Dear friends!In this article we will show you how to extract the root of a large number without a calculator. This is necessary not only to solve certain types Unified State Exam problems(there are some - for movement), but also for general mathematical development this analytical technique It is desirable to know.

It would seem that everything is simple: factor it into factors and extract it. No problem. For example, the number 291600 when decomposed will give the product:

We calculate:

There is one BUT! The method is good if divisors 2, 3, 4, and so on are easily determined. What should we do if the number from which we are extracting the root is a product prime numbers? For example, 152881 is the product of the numbers 17, 17, 23, 23. Try to find these divisors right away.

The essence of the method we are considering- This pure analysis. With developed skill, the root can be found quickly. If the skill has not been practiced, but the approach is simply understood, then it is a little slower, but still determined.

Let's take the root of 190969.

First, let’s determine between which numbers (multiples of one hundred) our result lies.

Obviously, the result of the root of given number lies in the range from 400 to 500, because

400 2 =160000 and 500 2 =250000

Really:

in the middle, closer to 160,000 or 250,000?

The number 190969 is approximately in the middle, but still closer to 160000. We can conclude that the result of our root will be less than 450. Let's check:

Indeed, it is less than 450, since 190,969< 202 500.

Now let's check the number 440:

This means our result is less than 440, since 190 969 < 193 600.

Checking the number 430:

We found that the result given root lies in the range from 430 to 440.

The product of numbers with 1 or 9 at the end gives a number with 1 at the end. For example, 21 by 21 equals 441.

The product of numbers with 2 or 8 at the end gives a number with 4 at the end. For example, 18 by 18 equals 324.

The product of numbers with a 5 at the end gives a number with a 5 at the end. For example, 25 by 25 equals 625.

The product of numbers with 4 or 6 at the end gives a number with 6 at the end. For example, 26 by 26 equals 676.

The product of numbers with 3 or 7 at the end gives a number with 9 at the end. For example, 17 by 17 equals 289.

Since the number 190969 ends with the number 9, it is the product of either the number 433 or 437.

*Only they, when squared, can give 9 at the end.

We check:

This means the result of the root will be 437.

That is, we seem to have “found” the correct answer.

As you can see, the maximum that is required is to carry out 5 actions in a column. Perhaps you will hit the mark right away, or take just three steps. It all depends on how exactly you do it initial estimate numbers.

Extract the root of 148996 yourself

Such a discriminant is obtained in the problem:

The motor ship travels 336 km along the river to its destination and, after stopping, returns to its point of departure. Find the speed of the ship in still water if the current speed is 5 km/h, the stay lasts 10 hours, and the ship returns to its point of departure 48 hours after departure. Give your answer in km/h.

View solution

The result of the root is between the numbers 300 and 400:

300 2 =90000 400 2 =160000

Indeed, 90000<148996<160000.

The essence of further reasoning comes down to determining how the number 148996 is located (distanced) relative to these numbers.

Let's calculate the differences 148996 - 90000=58996 and 160000 - 148996=11004.

It turns out that 148996 is close (much closer) to 160000. Therefore, the result of the root will definitely be greater than 350 and even 360.

We can conclude that our result is greater than 370. Further it is clear: since 148996 ends with the number 6, this means that we must square a number ending in either 4 or 6. *Only these numbers, when squared, give end 6.

Sincerely, Alexander Krutitskikh.

P.S: I would be grateful if you tell me about the site on social networks.

In mathematics, the question of how to extract a root is considered relatively simple. If we square numbers from the natural series: 1, 2, 3, 4, 5...n, then we get the following series of squares: 1, 4, 9, 16...n 2. The row of squares is infinite, and if you look closely at it, you will see that there are not very many integers in it. Why this is so will be explained a little later.

Root of a number: calculation rules and examples

So, we squared the number 2, that is, multiplied it by itself and got 4. How to extract the root of the number 4? Let's say right away that the roots can be square, cubic and any degree to infinity.

The power of the root is always a natural number, that is, it is impossible to solve the following equation: a root to the power of 3.6 of n.

Square root

Let's return to the question of how to extract the square root of 4. Since we squared the number 2, we will also extract the square root. In order to correctly extract the root of 4, you just need to choose the right number that, when squared, would give the number 4. And this, of course, is 2. Look at the example:

• 2 2 =4
• Root of 4 = 2

This example is quite simple. Let's try to extract the square root of 64. What number, when multiplied by itself, gives 64? Obviously it's 8.

• 8 2 =64
• Root of 64=8

Cube root

As was said above, roots are not only square; using an example, we will try to explain more clearly how to extract a cube root or a root of the third degree. The principle of extracting a cube root is the same as that of a square root, the only difference is that the required number was initially multiplied by itself not once, but twice. That is, let's say we took the following example:

• 3x3x3=27
• Naturally, the cube root of 27 is three:
• Root 3 of 27 = 3

Let's say you need to find the cube root of 64. To solve this equation, it is enough to find a number that, when raised to the third power, would give 64.

• 4 3 =64
• Root 3 of 64 = 4

Extract the root of a number on a calculator

Of course, it is best to learn to extract square, cube and other roots through practice, by solving many examples and memorizing tables of squares and cubes of small numbers. In the future, this will greatly facilitate and reduce the time required to solve equations. Although, it should be noted that sometimes you need to extract the root of such a large number that choosing the correct squared number will cost a lot of work, if possible at all. A regular calculator will come to the rescue in extracting the square root. How to extract the root on a calculator? Very simply enter the number from which you want to find the result. Now take a close look at the calculator buttons. Even the simplest of them has a key with a root icon. By clicking on it, you will immediately get the finished result.

Not every number can have a whole root; consider the following example:

Root of 1859 = 43.116122…

You can simultaneously try to solve this example on a calculator. As you can see, the resulting number is not an integer; moreover, the set of digits after the decimal point is not finite. Special engineering calculators can give a more accurate result, but the full result simply does not fit on the display of ordinary ones. And if you continue the series of squares that you started earlier, you will not find the number 1859 in it precisely because the number that was squared to obtain it is not an integer.

If you need to extract the third root on a simple calculator, then you need to double-click on the button with the root sign. For example, take the number 1859 used above and take the cube root from it:

Root 3 of 1859 = 6.5662867…

That is, if the number 6.5662867... is raised to the third power, then we get approximately 1859. Thus, extracting roots from numbers is not difficult, you just need to remember the above algorithms.

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 begin to re-solve the entire example. Under no circumstances should you do this! There are two reasons for this:

1. Roots of large numbers do appear in problems. Especially in text ones;
2. There is an algorithm by which these roots are calculated almost orally.

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

So, the algorithm:

1. Limit the required root above and below to numbers that are 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. The one whose square is equal to the original number will be the root.

Before putting this algorithm into practice, let's look at each individual step.

Root limitation

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

[Caption for the picture]

The same thing applies to 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 search area, move on to the second step.

Eliminating obviously unnecessary numbers

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

Believe it or not, we will now reduce the number of candidate numbers to two - 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, just 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 come in last place. Let's try to find out what they turn into when 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 relative 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. This means that, for example, the root of 3364 must end in 2 or 8. On the other hand, we remember the restriction from the previous paragraph. We get:

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:

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 there will be only one candidate for the roots!

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 gives the original number will be the root.

For example, for the number 3364 we found two candidate numbers: 52 and 58. Let's 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 for the squares of the sum and difference. Thanks to this, I didn’t even have to multiply the numbers into a column! This is another level of calculation optimization, but, of course, it is completely optional :)

Examples of calculating roots

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

[Caption for the picture]

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:

All that remains is to square each number and compare it with the original:

24 2 = (20 + 4) 2 = 576

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

Task. Calculate the square root:

[Caption for the picture]

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

Let's look at the last digit:

1369 → 9;
33; 37.

Square it:

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

Here is the answer: 37.

Task. Calculate the square root:

[Caption for the picture]

We limit the number:

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

Let's look at the last digit:

2704 → 4;
52; 58.

Square it:

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

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

Task. Calculate the square root:

[Caption for the picture]

We limit the number:

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

Let's look at the last digit:

4225 → 5;
65.

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;

Everything is correct. We write down the answer.

Conclusion

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

• In any normal mathematics exam, be it the State Examination or the Unified State Exam, the use of calculators is prohibited. And if you bring a calculator into class, you 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 when they see fractions, they generally become hysterical.