Question

Find the square root of each of the following numbers by using the method of prime factorisation:
(1) 225
(2) 441
(3) 729
(4) 1296
(5) 2025
(6) 4096
(7) 7056
(8) 8100
(9) 9216
(10) 11025
(11) 15876
(12) 17424

Answer 1

(1) 225= $3\times 3\times 5\times 5$

$\therefore \sqrt{225}=\sqrt{\overline{3\times 3}\times \overline{5\times 5}}=3\times 5=15$
(2) 441= $3\times 3\times 7\times 7$

$\therefore \sqrt{441}=\sqrt{\overline{3\times 3}\times \overline{7\times 7}}=3\times 7=21$
(3) 729= $\sqrt{729}=\sqrt{\overline{3\times 3}\times \overline{3\times 3}\times \overline{3\times 3}}$

$=3\times 3\times 3=27$
(4) 1296= $2\times 2\times 2\times 2\times 3\times 3\times 3\times 3$

$\therefore \sqrt{1296}=\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times \overline{3\times 3}\times \overline{3\times 3}}$
= $2\times 2\times 3\times 3=36$
(5) 2025 = $\sqrt{\overline{3\times 3}\times \overline{3\times 3}\times \overline{5\times 5}}$

=$3\times 3\times 5$
(6) $\sqrt{4096}$ = $\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times }\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}$

=$2\times 2\times 2\times 2\times 2\times 2$
(7) 7056= $2\times 2\times 2\times 2\times 3\times 3\times 7\times 7$

$\therefore \sqrt{7056}=\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times \overline{3\times 3}\times \overline{7\times 7}}$
= $2\times 2\times 3\times 7$
(8) $\sqrt{8100}=\sqrt{\overline{2\times 2}\times \overline{3\times 3}\times \overline{3\times 3}\times \overline{5\times 5}}$

= $2\times 3\times 3\times 5=90$
(9) $\sqrt{9216}=\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{2\times 2}\times \overline{3\times 3}}$

=$2\times 2\times 2\times 2\times 2\times 3=96$
(10) 11025= $3\times 3\times 5\times 5\times 7\times 7$
$\therefore \sqrt{11025}=\sqrt{\overline{3\times 3}\times \overline{5\times 5}\times \overline{7\times 7}}$

=$3\times 5\times 7=105$
(11) $\sqrt{15876}=\sqrt{\overline{2\times 2}\times \overline{3\times 3}\times \overline{3\times 3}\times \overline{7\times 7}}$

$=2\times 3\times 3\times 7=126$
(12) $sqrt17424=\sqrt{\overline{2\times 2}\times \overline{2\times 2}\times \overline{3\times 3}\times \overline{11\times 11}}$

=$2\times 2\times 3\times 11=132$