Square Root of Numbers

Given the square of these numbers find the square root:

a. 4 × 4  =  16, The square root of 16 is  written as  \(\scriptsize \sqrt{16}= 4\)
b. 5 × 5  =  25, The square root of 25, \(\scriptsize \sqrt{25}= 5\)
c. 3 × 3  = 9, The square root of 9,   \(\scriptsize \sqrt{9}= 3\)

This implies that the square root of a given number is that number that multiplies itself to give the given number.

Similarly,

d. -4 × -4 = 16, The square root of \(\scriptsize \sqrt{16}= 4\)
e. -5 × -5  =  25,   The square root of 25, \(\scriptsize \sqrt{25}= 5\)

In general, a number has two square roots: one negative square root and one positive square root.

e.g. \(\scriptsize \sqrt{16} \\ \scriptsize = \sqrt{4 \: \times \: 4}\\ \scriptsize = \sqrt{-4 \: \times \: -4} \\ \scriptsize = \pm{4}\)

\(\scriptsize \sqrt{25} \\ \scriptsize = \sqrt{5 \: \times \: 5}\\ \scriptsize = \sqrt{-5 \: \times \: -5}\\ \scriptsize = \pm{5} \)

Worked Example 2.2.1:

Find the square root of 

a. 144
b. 81
c. 324
d. 900

Solution

a. Square root of 144 = \(\scriptsize \sqrt{144}\)

Step 1: Express 144 as product of prime factors. 

Step 2: 

144  = (2 × 2) × (2 × 2) × (3 × 3)

\(\scriptsize \sqrt{144}\)   = 2 × 2 × 3

\(\scriptsize \sqrt{144}\)   = 12

b. Square  root of 81  =   \(\scriptsize \sqrt{81}\) 

Step 1: 

 81 = 3 × 3 × 3 × 3 

Step 2:

\(\scriptsize \sqrt{81}\)  = 3 × 3 = 9

c. Square root of  324 =  \(\scriptsize \sqrt{324}\)

Step 1:

324 = 2 × 2 × 3 × 3 × 3 × 3

Step 2:

\(\scriptsize \sqrt{324}\) = 2 × 3 × 3 = 18

d. Square root of 900   =  \(\scriptsize \sqrt{900}\)  

Step 1:

Step 2:

\(\scriptsize \sqrt{900}\)  = 2 × 3 × 5

= 30