Question

simlpify the following expression

Answer 1

Hey friend,
Here is the answer you were looking for:
${( \frac{81}{16} )}^{ - \frac{3}{4} } \times {( \frac{9}{25} )}^{ \frac{3}{2} } \div {( \frac{5}{2} )}^{ - 3}$

Splitting we get,

$= {( \frac{3 \times 3 \times 3 \times 3}{2 \times 2 \times 2 \times 2}) }^{ - \frac{3}{4} } \times (\frac{3 \times 3}{5 \times 5} ) ^{ \frac{3}{2} } \div (\frac{5}{2} )^{ - 3}$

We know that,
${( \frac{a}{b} )}^{ - m} = {( \frac{b}{a} )}^{m}$
${( \frac{ {3}^{4} }{ {2}^{4} }) }^{ - \frac{3}{4} } \times (\frac{ {3}^{2} }{ {5}^{2} })^{ \frac{3}{2} } \div ( \frac{2}{5} ) ^{3} \\ \\ \\ = {( \frac{3}{2} )}^{4 \times - \frac{3}{4} } \times {( \frac{3}{5} )}^{2 \times \frac{3}{2} } \times {( \frac{5}{2} )}^{3} \\ \\ \\ = {( \frac{3}{2} )}^{ - 3} \times {( \frac{3}{5} )}^{3} \times {( \frac{5}{2} )}^{3} \\ \\ \\ = {( \frac{2}{3} )}^{3} \times {( \frac{3}{5} )}^{3} \times {( \frac{5}{2} )}^{3} \\ \\ \\ = \frac{ {2}^{3} \times {3}^{3} \times {5}^{3} }{ {3}^{3} \times {5}^{3} \times {2}^{3} } \\ \\ using \: the \: identity \\ \\ \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n} \\ \\ \\ = {2}^{3 - 3} \times {3}^{3 - 3} \times {5}^{3 - 3} \\ \\ = {2}^{0} \times {3}^{0} \times {5}^{0} \\ \\ = 1 \times 1 \times 1 \\ \\ = 1$


Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
☺☺