Question

Find the square root of the following number by prime factorisation method
(i) $729$
(ii) $400$
(iii) $1764$
(iv) $4096$
(v) $7744$
(vi) $9604$
(vii) $5929$
(viii) $9216$
(ix) $529$
(x) $8100$

Answer 1

(i) $729$

$\begin{array}{cc}3& 729\\ 3& 243\\ 3& 81\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$\Rightarrow 729=3\times 3\times 3\times 3\times 3\times 3=$ $3^2\times 3^2\times 3^2$
$\therefore \sqrt{729}=27$
(ii) $400$

$\begin{array}{cc}2& 400\\ 2& 200\\ 2& 100\\ 2& 50\\ 5& 25\\ 5& 5\\ & 1\end{array}$

$\Rightarrow 400=2^2\times 2^2\times 5^2$
$\therefore \sqrt{400}=20$
(iii) $1764$

$\begin{array}{cc}2& 1764\\ 2& 882\\ 3& 441\\ 3& 147\\ 7& 49\\ 7& 7\\ & 1\end{array}$

$\Rightarrow 1764=2^2\times 3^2\times 7^2$
$\therefore \sqrt{1764}=42$
(iv) $4096$

$\begin{array}{cc}2& 4096\\ 2& 2048\\ 2& 1024\\ 2& 512\\ 2& 256\\ 2& 128\\ 2& 64\\ 2& 32\\ 2& 16\\ 2& 8\\ 2& 4\\ 2& 2\\ & 1\end{array}$

$\Rightarrow 1764=2^2\times 2^2\times 2^2\times 2^2\times 2^2\times 2^2$
$\therefore \sqrt{4096}=64$
(v) $7744$

$\begin{array}{cc}2& 7744\\ 2& 3872\\ 2& 1936\\ 2& 968\\ 2& 484\\ 2& 242\\ 11& 121\\ 11& 11\\ & 1\end{array}$

$\Rightarrow 7744=2^2\times 2^2\times 2^2\times {11}^2$
$\therefore \sqrt{7744}=88$
(vi) $9604$

$\begin{array}{cc}2& 9604\\ 2& 4802\\ 7& 2401\\ 7& 343\\ 7& 49\\ 7& 7\\ & 1\end{array}$

$\Rightarrow 9604=2^2\times 7^2\times 7^2$

$\therefore \sqrt{9604}=98$
(vii) $5929$

$\begin{array}{cc}7& 5929\\ 7& 847\\ 11& 121\\ 11& 11\\ & 1\end{array}$

$\Rightarrow 5929={11}^2\times 7^2$
$\sqrt{5929}=77$
(viii) $9216$

$\begin{array}{cc}2& 9216\\ 2& 4608\\ 2& 2304\\ 2& 1152\\ 2& 576\\ 2& 288\\ 2& 144\\ 2& 72\\ 2& 36\\ 2& 18\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$\Rightarrow 9216=2^2\times 2^2\times 2^2\times 2^2\times 2^2\times 3^2$

$\Rightarrow 9216={96}^2$
$\therefore \sqrt{9216}=96$
(ix) $529$

$\begin{array}{cc}23& 529\\ 23& 23\\ & 1\end{array}$

$\Rightarrow 529=23\times 23={23}^2$
$\therefore \sqrt{529}=23$
(x) $8100$

$\begin{array}{cc}2& 8100\\ 2& 4050\\ 5& 2025\\ 5& 405\\ 3& 81\\ 3& 27\\ 3& 9\\ 3& 3\\ & 1\end{array}$

$\Rightarrow 8100=2^2\times 3^2\times 3^2\times 5^2$

$\Rightarrow 8100={90}^2$

$\therefore \sqrt{8100}=90$.