What is the value of a given mathematical expression?
${(\frac{3}{5})}^{(4)} \cdot {(\frac{8}{5})}^{(- 12)} \cdot {(\frac{32}{5})}^{(6)}$

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\frac{2025}{64}

\frac{64}{2025}

$\frac{9}{125}$

\frac{125}{9}

answer is A.

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Detailed Solution

It is given that the mathematical expression is ${(\frac{3}{5})}^{(4)} \cdot {(\frac{8}{5})}^{(- 12)} \cdot {(\frac{32}{5})}^{(6)}$ .
The numerator contains 8 and 32 as bases, which are multiples of 2.
Hence, make the base as 2 and then solve using the laws of exponents.

$= {(\frac{3}{5})}^{(4)} \cdot {(\frac{2^{3}}{5})}^{(- 12)} \cdot {(\frac{2^{5}}{5})}^{(6)}$

$= (\frac{3^{4}}{5^{4}}) \cdot (\frac{2^{(- 36)}}{5^{(- 12)}}) \cdot (\frac{2^{30}}{5^{6}})$

$= (\frac{3^{4}}{5^{4}}) \cdot (\frac{5^{12}}{2^{36}}) \cdot (\frac{2^{30}}{5^{6}})$
$\begin{array}{l} = \frac{3^{4} \times 5^{12 - 4 - 6}}{2^{36 - 30}} \\ = \frac{3^{4}}{2^{6}} \times 5^{2} \\ = \frac{81 \times 25}{64} \\ = \frac{2025}{64} \end{array}$
Therefore, the correct answer is $\frac{2025}{64}$ .