How To Calculate A Square Root

Welcome to this article where I will give you all the necessary tools to learn how to calculate the square root of a number. In order to understand this, you must already have a good understanding of addition and subtraction, and above all, have notions of multiplication and division.

If you have knowledge about these basic things, then you can continue reading to learn how to calculate the square root of a number, be it a whole number, natural numbers, decimals, and fractions. Something you should know is that looking for the square root of a number means that we are going to find what number multiplied by itself, or raised to the square, gives us the original number as a result, that is, the number to which we want to find the root.

For example: in the case of integers, if what you need is to take the square root of 16, we must reason that if 4×4 is equal to 16. Then the square root of 16 must be 4. The same thing happens with all integers. If what we are trying to do is find the square root of 36 then we are going to reason: If 6 × 6 is equal to 36, then the square root of 36 must be 6.

As you are surely realizing by now, calculating the square root of a number is quite simple once you apply some logic to its concept. Basically, finding the square root of a number means that we are going to look for the number raised to the square or multiplied by itself, which gives us the number with which we started.

Now let’s think calmly about this: We already know that there are numbers whose square root is exact, whether within the natural, decimal or fractional numbers, but there are also other numbers that are not exact and to be able to calculate their square root well, it requires applying a special algorithm in order to obtain the accuracy of the number or obtain some approximation, depending on what you are looking for.

If so far you have not missed the explanation, continue reading. We then know that numbers whose square roots are exact are called quadratic numbers. These quadratic numbers come from raising each number by itself, or in other words squared. When we say that we are going to square a number, basically what we are doing is multiplying it right away. Therefore, if we want to square 2: 2² we are going to multiply two by itself, that is, 2×2. Therefore 2 ² =4.

We highlight this to be able to affirm then that when searching for example, the square root of an integer and natural number such as 49, what we will be looking for is a number that squared gives us 49 as a result. And what number is that? 7

7×7= 49

7 ² = 49

Square root of 49 = 7

To calculate non-exact roots, we are going to need to implement other methods and all this can be calculated very easily with a scientific calculator, however if you have come this far it is because you want to learn how to calculate square curl without necessarily using a calculator. That is why I will give you some instructions that you can take into account.

What do you need to calculate a square root?

• Math book
• Notebook

Instructions for calculating a square root

To calculate a square root we must also know the parts that make it up. That is to say: the number that is inside the symbol of the root √ 121 is called the radicand.

The small number above the top of the root symbol is called the index. In this example: √121 the index is not drawn therefore whenever the index is not drawn before the root, it means that we will take the square root of the radicand.

We can find situations in which the index appears drawn and situations in which it is not.  As long as it is not drawn, we must know that it is because we have to look for the square root, if for example there were a 3 in front: 3 √121 means that we will be looking for the cube root.

Examples:

0 ² = √0 = 0

1 ² = √1 = 1

2 ² = √4 = 2

3 ² = √9 = 3

4 ² = √16 = 4

5 ² = √25 = 5

6 ² = √36 = 6

7 ² = √49 = 7

8 ² = √64 = 8

9 ² = √81 = 9

10 ² = √100 = 10

11 ² = √121 = 11

12 ² = √144 = 12

In the case of integers, it is a matter of memorizing some numbers to know how to quickly take their square root. Numbers such as 20, 30, 40, 100 or 1000, since it is easy to multiply them by their same number, it is also easy to take their square root.

For example: if we know 20×20 is equal to 400. We can logically assume that the square root of 400 is 20.

The real problem of taking the square root of a number is in the event that the result does not give us another whole number, but a decimal number, as it can be in the case of wanting to take the square root of 500.

See the following example:

10×10= 100 so √100 = 10

Instead √110 = 10.4880884817

In these cases, what we want to do is take the number that is approximate to the square root of a given number. As it can be, for example, in the case that we are given to calculate the square root of 30.

We know in this case that the square root of 25 is 5 and we also know that the square root of 36 is 6 then. Why do we take these two results into account?

It is simple, since we want to take the square root of 30, logically we know that it will be a number between the aforementioned integers 5 and 6

Therefore, √30 = 5,…..

The ideal in these cases is that you calculate the square root with a scientific calculator, since it will make your job much easier. But if what you want is to learn how to take the square root manually, keep reading the following example and pay close attention:

• √89224

If the radicand is made up of several numbers, what you will do is divide the numbers in the radicand into pairs starting from the decimal to the hundredths. That is, from right to left:

 √8 92 24

• First we will calculate the square root of the first number. That is to say, in this case, the number 8. We will find that 8 is not a perfect square, since it is located between two perfect squares, which in this case are 4 and 9.

4 is the square root of 2

And 9 is the square root of 3.

What we have to do in this case is place the number 2 since this is the smallest and it does not exceed the number that they offer us: in this case 8.

√8 92 24 | 2

• Because the root number obtained, that is, 4 is less than 8, we will place the number 4 below 8 to proceed to subtract it as follows:

√8 92 24 | 2

-4 | 2×2= 4

___

4

• Now we will proceed to “lower” the next pair of numbers. Which in this case is 92, as follows:

√8 92 24 | 2

-4 | 2×2=4

___

4 92

• We separate the numbers into pairs again. We will have 49 left on one side and 2 on the other. And we will proceed to divide 49 by twice what we obtained in the last root. That is, since in the previous root we obtained 2, we will divide 49 by 4 as follows:

49%4=12.25 > 9

What we will be left with is something similar to this:

√8 92 24 | 2

-4 | 2×2=4

___

49 2 | 49×9= 441

6) Now we will continue with the subtraction. From the number we had of 492, we will subtract the result of the multiplication of 49×9, that is, 441

√8 92 24 | 2

-4 | 2×2=4

___

492 | 49 x 9 = 441

-441

___

51

If the result between 49×9 would have given us a number greater than 492, we would have to multiply 48 by a smaller number, either 8 or 7, until obtaining a number that was less than 492. In such a way that we could perform the subtraction.

Now if we can bring down the last pair of numbers, which in this case is 24.

We will have something like this

√8 92 24 | 2

-4 | 2×2=4

___

492 | 49 x 9 = 441

-441

___

512 4

• Now we will repeat the logic applied previously.

Because 589×9 gave us a result that exceeded the number from which we then have to subtract, that is, it exceeded 5124, we had to multiply 589×8 instead of x9

√8 92 24 | 29

-4 | 2×2=4

___

492 | 49×9=441

-441

___

512 4 | 589 x 8 = 4704

Now we will continue subtracting in this way

√8 92 24 | 298

-4 | 2×2=4

___

492 | 49×9=441

-441

___

512 4 | 589×8=4704

-4704

_____

420

• Finally, as a result we can say that √89224 = 298 ² + 420

Tips for Calculating a Square Root

1. Use a scientific calculator to avoid complicating your life when doing this algorithm.
2. Remember that the square root is not written over the root. As long as the root is empty, it means that you will have to find the square root to get the result.
3. It is recommended to practice a lot and hard so that this account comes out easily.