Square Root Finder with Step by Step Solutions

We calculate the area of a square as a product of side i.e side², but if we have to calculate the side of a square we need to take the square root of the area. Thus we can say that the square root is the inverse operation of squaring a number. Square roots are used in solving algebraic equations and other complex Mathematical and Scientific calculations.Square root of a number has two values, positive and negative. Now let us study root finder methods to find the square root of imperfect squares.

What are Square Roots?

The square root of a number is that number which when multiplied by itself gives the number itself whose square root has to be found. The number x is a perfect square. Finding the square root of a number is equivalent to raising the same number to the power ½. The square root of a number ‘x’ can therefore be written in exponential form as x\[^{\frac{1}{2}}\]. It is denoted by the symbol √. The square root of a number x is written a √x or x\[^{\frac{1}{2}}\]. Any number when multiplied by itself gives the square of a number. For Example, the square root of 4 is 2, the square root of 9 is 3 and the square root of 16 is 4. All these are perfect squares so we can easily find out the square root. But if the numbers are imperfect squares then how to find the square root of imperfect numbers. In this article let us study how to calculate imperfect square roots and different root finder techniques.

Finding Square Roots:

First, check whether the given number is a perfect square number or not. If it is a product of any number by itself then it is a perfect square To find the square root of perfect square numbers, any one of the following methods can be used. 

• Prime factorization method

• Repeated subtraction method

• Long division method

• Number line method

• Average method

But, if the number is not a perfect square prime factorization method and the repeated subtraction method will not work, we have to use other methods for finding the square roots. Let us study how to calculate imperfect square roots.

Finding Square Roots of Imperfect Squares

How to find the square root of imperfect numbers by the average method: 

We will use the average method to find out the square root of an imperfect square. Let us find the square root 5 using the average method, the following are the steps.

Step 1: Find out the two perfect square numbers which are very close to the given number on either side. For example the number ‘10’, the immediate perfect square lesser than 10 is ‘9’ and the immediate perfect square greater than 10 is ‘16’.

Step 2 : Note down the square roots of the perfect squares, here the square root of ‘9’ is ‘3’ and the square root of ‘16’ is ‘4’.

Step 3 : Square root of a given number lies between the square roots of numbers determined in step 2.Square root of ‘10’ is any number between 3 and 4.

Step 4: Divide the number whose square root is determined by any of the numbers obtained in Step 2.‘10’ can be divided either by ‘3’ or ‘4’.

Let us divide ‘10’ by ‘3’

\[\frac{10}{3}\] = 3.33

Step 5: Find the average of the quotient and divisor in Step 4.

The average of 3 and 3.33 is

Average = \[\frac{3+3.33}{2}\] = \[\frac{6.33}{2}\] = 3.165

Step 6: Now divide 10 by step 5 answer

\[\frac{10}{3.165}\] = 3.159

Step 7: Now, average 3.1579 and 3.1667 by adding them together and dividing the sum  by two you get 3.1623. Check your work by multiplying your answer by itself. If 3.1623 is multiplied by 3.1623 we get 10.001.

Therefore \[\sqrt{10}\] = 3.1623

How to find the square root of imperfect numbers by long division method

Let us find the square root of 104976 by using a long division method. 

Step 1 :

Separate the numbers by taking commas from right to left in a group of two digits.

Such as 

10,49,76  

Step 2 :

Now we have to multiply a number by itself such that the product is less than or equal to 10

here, 3 x 3  =  9  will meet the condition

Now, 9 is subtracted from 10 and we got the remainder 1.

Step 3 :

Now, we have to bring down 49 so the remainder becomes 149 and add 3 to quotient 3 we get 6.

Step 4 :

Then, we have to find a product such that it is less than 149 and the same number will be written in the divisor column in quotient.

Step 5 :

The condition said in step 4 will be met by "2". 62 multiplied by 2 we get 124. We get remainder 25.

Step 6 :

Now, we have to bring down 76 and add 2 to 62. We get the remainder of 2576 and divisor 64.

Step 7 :

Now multiply 644 by 4 we get 2576. So the remainder becomes zero.

Step 8 :

Finally, we got the square root of 104976  is 324.

Below figure represents the long division method. From these steps it is clear how to calculate imperfect square roots.

Properties of Square Root

• Only a perfect square number has a perfect square root.

• The square root of an even perfect square is even.

• Because a perfect square cannot be negative, the square root of a negative number cannot be defined.

• A number cannot have a square root if it finishes with an odd number of zeros. A square root can only be calculated with an even number of zeros.

• In a set of real numbers, negative values have no square roots.

• A square root is found in numbers that conclude with (containing a unit's digit) 1, 4, 5, 6, or 9.