How to calculate the square root of 100. Square root

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:

Roots of large numbers do appear in problems. Especially in text ones;
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:

Limit the required root above and below to numbers that are multiples of 10. Thus, we will reduce the search range to 10 numbers;
From these 10 numbers, weed out those that definitely cannot be roots. As a result, 1-2 numbers will remain;
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:

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The same thing applies to any other number from which you can find the square root. For example, 3364:

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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'll 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:

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

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

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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:

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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.

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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:
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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:

At any normal exam in mathematics, 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.

The problem of finding a root in mathematics is the inverse problem of raising a number to a power. There are different roots: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is indicated by the symbol: √ is a square root, that is, the root of the second degree; if the root has a degree greater than the second, then the corresponding degree is assigned above the root sign. The number that is under the root sign is a radical expression. When finding a root, there are several rules that will help you not make a mistake in finding the root:

An even root (if the degree is 2, 4, 6, 8, etc.) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree is sought (3, 5, 7, and so on), then the result will be negative.
The root of any power of one is always one: √1 = 1.
The root of zero is zero: √0 = 0.

How to find the root of 100

If the problem does not say what root of the degree needs to be found, then it usually means that it is necessary to find the root of the second degree (square).
Let's find √100 = ? We need to find a number that, when raised to the second power, gives the number 100. Obviously, such a number is the number 10, since: 10 2 = 100. Therefore, √100 = 10: the square root of 100 is 10.

Among the many knowledge that is a sign of literacy, the alphabet comes first. The next, equally “sign” element is the skills of addition-multiplication and, adjacent to them, but opposite in meaning, arithmetic operations of subtraction-division. The skills learned in distant school childhood serve faithfully day and night: TV, newspaper, SMS, and everywhere we read, write, count, add, subtract, multiply. And, tell me, have you often had to extract roots in your life, except at the dacha? For example, such an entertaining problem, like the square root of the number 12345... Is there still gunpowder in the flasks? Can we handle it? Nothing could be simpler! Where is my calculator... And without it, hand-to-hand combat is weak?

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

Taking the square root

From the school arithmetic course, methods of calculations “in a column” are known, which help to perform any calculations using the first four arithmetic operations. Alas... For square, and not only square, roots, such algorithms do not exist. And in this case, how to extract the square root without a calculator? Based on the definition of the square root, there is only one conclusion - it is necessary to select the value of the result by sequentially enumerating numbers whose square approaches the value of the radical expression. That's all! Before an hour or two has passed, you can calculate, using the well-known method of multiplication in a “column”, any square root. If you have the skills, this will only take a couple of minutes. Even a not-so-advanced user of a calculator or PC can do this in one fell swoop - progress.

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

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

1. Let's take, purely intuitively, X=100. Let's calculate: X * X = 10000. Intuition is at its best - the result is less than 12345.

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

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

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

Just a little history...

The Pythagoreans, students of the school and followers of Pythagoras, came up with the idea of using square roots, 800 years BC. and then we “ran into” new discoveries in the field of numbers. And where did that come from?

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

2. It is known that it turned out that this mathematical operation contains another catch - when extracting the root, we do not know which number, positive or negative, is the square of the radical expression. This uncertainty, the double result from one operation, is recorded in this way.

The study of problems related to this phenomenon has become a direction in mathematics called the theory of complex variables, which has great practical importance in mathematical physics.

It is curious that the same ubiquitous I. Newton used the designation of the root - radical - in his “Universal Arithmetic”, and exactly the modern form of notation of the root has been known since 1690 from the book of the Frenchman Rolle “Manual of Algebra”.

Today we will figure out on this page of our website what the square root of 100 is. Let's figure out together what the square root of 100 is, since 1000 scientists have been racking their brains on this topic for many decades, and many have come to the inevitable conclusion from calculations that such a root does not exist at all and it is simply impossible to calculate it. It is also very important in this case to ask exactly the right question to identify the square root of 100. To be precise, we will calculate the arithmetic square root of 100, since in the ordinary square root of 100 we will end up with two numbers: 10 and - 10.

We can calculate the sum of these numbers we need using a simple arithmetic technique using a vertical, familiar line, numbers and roots that are written in the lower right. There we will find the square of units of the root we need, then multiply the tens and find the double and not triple the product of the ten of any root by units. We will have to square some numbers so that the total becomes a two-digit number; if in the end we get the number 10, then we have done everything right with you. The main thing is to initially become at least a little familiar with mathematics and the mathematical progression of composing the square root before starting calculations.

Remember one single and basic rule: in order to extract the necessary square root from any integer, first of all we extract any root we need from the number of its sums and hundreds. If the number is equal to or greater than 100, then we begin to look for the root of the hundreds of actual numbers of these hundreds, then of the tens of thousands of the actual number, especially if the given number is much more than 100, then we necessarily extract the root of the number of hundreds of tens of thousands or to be more precise: out of a million of a given number. There are many rules and various scientific recommendations on this topic; school programs for extracting the square root of the number 100 will always remain unchanged.

If we consider the progress of finding the root of the number 100, we need to pay attention to the fact that there are as many digits in the root as there are under a finite number of sides, while the left side can consist of only one digit. Based on all this, the most accurate square root of any number on planet earth will be the sum of numbers whose square is exactly equal to the given number when calculated. This is where we can finish our short course on calculating the square root of 100 which will be equal to (10) ten.

Konstantinova Vera

How to find the root of a number

The problem of finding a root in mathematics is the inverse problem of raising a number to a power. There are different roots: roots of the second degree, roots of the third degree, roots of the fourth degree, and so on. It depends on what power the number was originally raised to. The root is denoted by the symbol: √ is a square root, that is, the root of the second degree; if the root has a degree greater than the second, then the corresponding degree is assigned above the root sign. The number that is under the root sign is a radical expression. When finding a root, there are several rules that will help you not make a mistake in finding the root:

An even root (if the degree is 2, 4, 6, 8, etc.) of a negative number does NOT exist. If the radical expression is negative, but the root of an odd degree is sought (3, 5, 7, and so on), then the result will be negative.
The root of any power of one is always one: √1 = 1.
The root of zero is zero: √0 = 0.

How to find the root of 100

I looked again at the sign... And, let's go!

Let's start with something simple:

Just a minute. this, which means we can write it like this:

Got it? Here's the next one for you:

Are the roots of the resulting numbers not exactly extracted? No problem - here are some examples:

What if there are not two, but more multipliers? The same! The formula for multiplying roots works with any number of factors:

Now completely on your own:

Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!

Root division

We've sorted out the multiplication of roots, now let's move on to the property of division.

Let me remind you that the general formula looks like this:

Which means that the root of the quotient is equal to the quotient of the roots.

Well, let's look at some examples:

That's all science is. Here's an example:

Everything is not as smooth as in the first example, but, as you can see, there is nothing complicated.

What if you come across this expression:

You just need to apply the formula in the opposite direction:

And here's an example:

You may also come across this expression:

Everything is the same, only here you need to remember how to translate fractions (if you don’t remember, look at the topic and come back!). Do you remember? Now let's decide!

I am sure that you have coped with everything, now let’s try to raise the roots to degrees.

Exponentiation

What happens if the square root is squared? It's simple, remember the meaning of the square root of a number - this is a number whose square root is equal to.

So, if we square a number whose square root is equal, what do we get?

Well, of course, !

Let's look at examples:

It's simple, right? What if the root is to a different degree? It's OK!

Follow the same logic and remember the properties and possible actions with degrees.

Read the theory on the topic “” and everything will become extremely clear to you.

For example, here is an expression:

In this example, the degree is even, but what if it is odd? Again, apply the properties of powers and factor everything:

Everything seems clear with this, but how to extract the root of a number to a power? Here, for example, is this:

Pretty simple, right? What if the degree is more than two? We follow the same logic using the properties of degrees:

Well, is everything clear? Then solve the examples yourself:

And here are the answers:

Entering under the sign of the root

What haven’t we learned to do with roots! All that remains is to practice entering the number under the root sign!

It's really easy!

Let's say we have a number written down

What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!

Why do we need this? Yes, just to expand our capabilities when solving examples:

How do you like this property of roots? Does it make life much easier? For me, that's exactly right! Only We must remember that we can only enter positive numbers under the square root sign.

Solve this example yourself -
Did you manage? Let's see what you should get:

Well done! You managed to enter the number under the root sign! Let's move on to something equally important - let's look at how to compare numbers containing a square root!

Comparison of roots

Why do we need to learn to compare numbers that contain a square root?

Very simple. Often, in large and long expressions encountered in the exam, we receive an irrational answer (remember what this is? We already talked about this today!)

We need to place the received answers on the coordinate line, for example, to determine which interval is suitable for solving the equation. And here the problem arises: there is no calculator in the exam, and without it, how can you imagine which number is greater and which is less? That's it!

For example, determine which is greater: or?

You can’t tell right away. Well, let's use the disassembled property of entering a number under the root sign?

Then go ahead:

Well, obviously, the larger the number under the root sign, the larger the root itself!

Those. if, then, .

From this we firmly conclude that. And no one will convince us otherwise!

Extracting roots from large numbers

Before this, we entered a multiplier under the sign of the root, but how to remove it? You just need to factor it into factors and extract what you extract!

It was possible to take a different path and expand into other factors:

Not bad, right? Any of these approaches is correct, decide as you wish.

Factoring is very useful when solving such non-standard problems as this:

Let's not be afraid, but act! Let's decompose each factor under the root into separate factors:

Now try it yourself (without a calculator! It won’t be on the exam):

Is this the end? Let's not stop halfway!

That's all, it's not so scary, right?

Happened? Well done, that's right!

Now try this example:

But the example is a tough nut to crack, so you can’t immediately figure out how to approach it. But, of course, we can handle it.

Well, let's start factoring? Let us immediately note that you can divide a number by (remember the signs of divisibility):

Now, try it yourself (again, without a calculator!):

Well, did it work out? Well done, that's right!

Let's sum it up

The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal to.
.
If we simply take the square root of something, we always get one non-negative result.
Properties of an arithmetic root:
When comparing square roots, it is necessary to remember that the larger the number under the root sign, the larger the root itself.