Question
Simplify : $\frac{{{3^{ - 5}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}$
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Answer
$\frac{{{3^{ - 5}} \times {{10}^{ - 5}} \times 125}}{{{5^{ - 7}} \times {6^{ - 5}}}}$
$ = \frac{{{3^{ - 5}} \times {{(2 \times 5)}^{ - 5}} \times (5 \times 5 \times 5)}}{{{5^{ - 7}} \times {{(2 \times 3)}^{ - 5}}}}$
$ = \frac{{{3^{ - 5}} \times {2^{ - 5}} \times {5^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}$
$ = \frac{{{5^{ - 5}} \times {5^3}}}{{{5^{ - 7}}}}$
$ = \frac{{{5^{(5) + 3}}}}{{{5^{ - 7}}}}$
$ = \frac{{{5^{ - 2}}}}{{{5^{ - 7}}}}$
$ =5^{(-2)-(-7)} $
$ =5^{-2+7} $
$ =5^5 $
$ = \frac{{{3^{ - 5}} \times {{(2 \times 5)}^{ - 5}} \times (5 \times 5 \times 5)}}{{{5^{ - 7}} \times {{(2 \times 3)}^{ - 5}}}}$
$ = \frac{{{3^{ - 5}} \times {2^{ - 5}} \times {5^{ - 5}} \times {5^3}}}{{{5^{ - 7}} \times {2^{ - 5}} \times {3^{ - 5}}}}$
$ = \frac{{{5^{ - 5}} \times {5^3}}}{{{5^{ - 7}}}}$
$ = \frac{{{5^{(5) + 3}}}}{{{5^{ - 7}}}}$
$ = \frac{{{5^{ - 2}}}}{{{5^{ - 7}}}}$
$ =5^{(-2)-(-7)} $
$ =5^{-2+7} $
$ =5^5 $
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