Q.5 Express each of the following as the product of powers of their prime
(i) 648
(ii) 405
(iii) 540
(iv) 3600


Answer :

(i) $$ \begin{array}{|l} \llap{2~~~~} 648 \\ \hline \llap{2~~~~} 324 \\ \hline \llap{2~~~~} 162 \\ \hline \llap{3~~~~} 81 \\ \hline \llap{3~~~~} 27 \\ \hline \llap{3~~~~} 9 \\ \hline \llap{3~~~~} 3 \\ \hline 1 \end{array} $$ 648= \(2^3× 3^4 \)
(ii) $$ \begin{array}{|l} \llap{3~~~~} 405 \\ \hline \llap{3~~~~} 135 \\ \hline \llap{3~~~~} 45 \\ \hline \llap{3~~~~} 15 \\ \hline \llap{5~~~~} 5 \\ \hline \llap{~~~~} 1 \\ \hline \end{array} $$ 405= \(3^4×5^1 \)
(iii) $$ \begin{array}{|l} \llap{2~~~~} 540 \\ \hline \llap{2~~~~} 270 \\ \hline \llap{3~~~~} 135 \\ \hline \llap{3~~~~} 45 \\ \hline \llap{3~~~~} 15 \\ \hline \llap{5~~~~} 5 \\ \hline \llap{~~~~} 1 \\ \hline \end{array} $$ \( 540 = 2^2×3^3×5^1 \)
(iv) $$ \begin{array}{|l} \llap{2~~~~} 3600 \\ \hline \llap{2~~~~} 1800 \\ \hline \llap{2~~~~} 900 \\ \hline \llap{2~~~~} 450 \\ \hline \llap{3~~~~} 225 \\ \hline \llap{3~~~~} 75 \\ \hline \llap{5~~~~} 25 \\ \hline \llap{5~~~~} 5 \\ \hline 1 \end{array} $$ \(3600 = 2^4×3^2×5^2 \)

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