• # question_answer 12) Express each of the following as a product of prime factors only in exponential form: (i) $108\times 192$                       (ii) $\text{27}0$                                              (iii) $729\times 64$                       (iv) $768$

(i) $\mathbf{108\times 192}$                 $108\times 192$$=(2\times 2\times 3\times 3\times 3)\times (2\times 2\times 2\times 2\times 2\times 2\times 3)$ $=2\times 2\times 3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2\times 3$ $=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3\times 3\times 3\times 3$ $={{2}^{8}}\times {{3}^{4}}$

 2 108 2 54 3 27 3 9 3 3 1
 2 192 2 96 2 48 2 24 2 12 2 6 3 3 1
it is the required prime factor product form                 (ii) 270                  2 270 3 135 3 45 3 15 5 5 1
$\therefore$  $270=2\times 3\times 3\times 3\times 5={{2}^{1}}\times {{3}^{3}}\times {{5}^{1}}$                 It is the required prime factor product form.                 (iii) $\mathbf{729\times 64}$                 $729\times 64=3\times 3\times 3\times 3\times 3\times 3\times 2\times 2\times 2\times 2\times 2\times 2$                  3 729 3 243 3 81 3 27 3 9 3 3 1
 2 64 2 32 2 16 2 8 2 4 2 2 1
$={{3}^{6}}\times {{2}^{6}}$                 It is the required prime factor product form.                 (iv) 768                  2 768 2 384 2 192 2 96 2 48 2 24 2 12 2 6 3 3 1
$\therefore$  $768=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 3$ $={{2}^{8}}\times {{3}^{1}}$     It is the required prime factor product form.

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