Question

For each of the following number, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.

$\left(i\right)252$ $\left(ii\right)180$ $\left(iii\right)1008$ $\left(iv\right)2028$ $\left(v\right)1458$ $\left(vi\right)768$

Answer 1

i) $252=\underset{–}{2\times 2}\times \underset{–}{3\times 3}\times 7$

Therefore, we need to multiply $252$ by $7$ in order to make it a perfect square.

So, $252\times 7=1764$

$\therefore \sqrt{1764}=42$

ii) $180=\underset{–}{2\times 2}\times \underset{–}{3\times 3}\times 5$

Therefore, we need to multiply $180$ by $5$ in order to make it a perfect square.

So, $180\times 5=900$

$\therefore \sqrt{900}=30$

iii) $1008=\underset{–}{2\times 2}\times \underset{–}{2\times 2}\times \underset{–}{3\times 3}\times 7$

Therefore, we need to multiply $1008$ by $7$ in order to make it a perfect square.

So, $1008\times 7=7056$

$\therefore \sqrt{7056}=84$

iv) $2028=\underset{–}{2\times 2}\times \underset{–}{13\times 13}\times 3$

Therefore, we need to multiply $2028$ by $3$ in order to make it a perfect square.

So, $2028\times 3=6084$

$\therefore \sqrt{6084}=78$

v) $1458=\underset{–}{3\times 3}\times \underset{–}{3\times 3}\times \underset{–}{3\times 3}\times 2$

Therefore, we need to multiply $1458$ by $2$ in order to make it a perfect square.

So, $1458\times 2=2916$

$\therefore \sqrt{2916}=54$

v) $768=\underset{–}{2\times 2}\times \underset{–}{2\times 2}\times \underset{–}{2\times 2}\times \underset{–}{2\times 2}\times 3$

Therefore, we need to multiply $768$ by $3$ in order to make it a perfect square.

So, $768\times 3=2304$

$\therefore \sqrt{2304}=48$