Question

Find the square roots of the following numbers by the 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) $729$ can be factorized as follows.

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

$\Rightarrow 729=3\times 3\times 3\times 3\times 3\times 3$

$=3^2\times 3^2\times 3^2$

$\therefore \sqrt{729}=3\times 3\times 3=27$

(ii) $400$ can be factorized as follows.

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

$\Rightarrow 400=2\times 2\times 2\times 2\times 5\times 5$

$=2^2\times 2^2\times 5^2$

$\therefore \sqrt{400}=2\times 2\times 5=20$

(iii) $17264$ can be factorized as follows.

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

$\Rightarrow 17264=2\times 2\times 3\times 3\times 7\times 7$

$=2^2\times 3^2\times 7^2$

$\therefore \sqrt{1764}=2\times 3\times 7=42$

(iv) $4096$ can be factorized as follows.

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

$\Rightarrow 4096=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$

$=2^2\times 2^2\times 2^2\times 2^2\times 2^2\times 2^2$

$\therefore \sqrt{1764}=2\times 2\times 2\times 2\times 2\times 2=64$

(v) $7744$ can be factorized as follows.

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

$\Rightarrow 7744=2\times 2\times 2\times 2\times 2\times 2\times 11\times 11$

$=2^2\times 2^2\times 2^2\times {11}^2$

$\therefore \sqrt{1764}=2\times 2\times 2\times 11=88$

(vi) $9604$ can be factorized as follows.

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

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

$=2^2\times 7^2\times 7^2$

$\therefore \sqrt{9604}=2\times 7\times 7=98$

(vii) $5929$ can be factorized as follows.

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

$\Rightarrow 5929=7\times 7\times 11\times 11$

$=7^2\times {11}^2$

$\therefore \sqrt{5929}=7\times 11=77$

(viii) $9216$ can be factorized as follows.

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

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

$=2^2\times 2^2\times 2^2\times 2^2\times 2^2\times 3^2$

$\therefore \sqrt{5929}=2\times 2\times 2\times 2\times 2\times 3=96$

(ix) $529$ can be factorized as follows.

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

$\Rightarrow 529=23\times 23$

$={23}^2$

$\Rightarrow \sqrt{529}=23$

(x) $8100$ can be factorized as follows.

$\begin{array}{cc}2& 8100\\ 2& 4050\\ 3& 2025\\ 3& 675\\ 3& 225\\ 3& 75\\ 5& 25\\ 5& 5\\ & 1\end{array}$

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

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

$\therefore \sqrt{529}=2\times 3\times 3\times 5=90$