Square root of 144 – value, derivation and solved examples

In mathematics, we can calculate the square root of a number by inverting the squaring operation. For instance, the value of the square root of 144 is 12 because 12 x 12 gives 144. We can also write this equation in radical form as √144 = 12. The preceding equation shows that the natural number 144 gives a square root of 12. This is a value that gives the number 144 when multiplied by itself.

Don’t worry; if you can’t remember this, we’ll share two methods to calculate the value of √144 – prime factorisation and long division.  In this article, we will learn both methods and look at some solved examples to strengthen your understanding of the concept of square roots.

What does the square root of a number mean?   

A value expressed by x = √a represents the square of any natural integer. It means that x, where a is any natural integer, is the square root of a. Thus, it is inferred that the square root of any number is identical to a figure that gives the original numbers when multiplied by itself. For instance, 4 x 4 = 16, and the root of the square of the number 16 may be stated to be 4. The symbol ‘√’ represents a square root and is also called ‘radical’.

How to find the value of the square root of 144 by prime factorisation?

To find the value of √144 by prime factorisation, follow the steps mentioned below:

1. First, calculate the prime factors of 144 and write them down.
   144 = 2 x 2 x 2 x 2 x 3 x 3
2. Group these prime factors in a pair of two factors each.
   144 = 2² x 2² x 3²
3. Add a square root on both sides.
   √144 = √(2² x 2² x 3²)
   √144 = √(2 x 2 x 3)²
   √144 = (2 x 2 x 3)^{2 x ½}
   √144 = 12

By following these steps, you can calculate the prime factors of √144 within no time.

Finding the value of √144 by long division

We may also use the long division approach to determine the square root of any number. This approach is highly beneficial and the fastest way to discover the root. The root of the imperfect squares and of big numbers cannot be found by prime factorisation, but long division comes to the rescue here. The instructions below are for applying long division to find the value of √144.

1. As indicated in the image, write 144. Start by placing a bar above it, grouping the number in pairs of two digits from the right. The left unpaired number can be regarded as a single entity. For instance, for the number 144, 44 has one bar above it, and 1 has the second.
2. We know that digit 1’s square root is always 1. Therefore, given that 1 is the quotient, dividend and divisor, the remainder equals 0.
3. Now, we’re going to bring down the next two numbers as a dividend, i.e. 44, and add 1 to our next divisor, i.e. 1+1 =2.
4. Select a number for the unit place of the divisor to yield 44 or a smaller number closest to 44 when multiplied by the new divisor. The number is 22 since 22 x2 = 44 here.

Solved examples

1. Find the square root of 81 by prime factorisation.

Follow the steps below to find the value of √81

• Calculate the prime factors of 81 and write them down.
  81 = 3 x 3 x 3 x 3
• Group these prime factors in a pair of two factors each.
  81 = 3² x 3²
• Add a square root on both sides
  √81 = √(3² x 3²)
  √81 = √(3 x 3)²
  √81 = (3 x 3)^{2 x ½}
  √81 = 9

So, √81 = 9.

2. Simplify √81 + √144 – √25

Given, √81 + √144 – √25

We know that √81 = 9, √144 = 12  and √25 = 5

Putting these square root values in the equation

We get, 9 + 12 – 5 = 16

So,  √81 + √144 – √25 upon simplification gives 16.