Question

simplify :

[81/16]^-3/4 into [25/9]^-3/2​

Answer 1

${( \frac{81}{16})}^{ \frac{ - 3}{4}} \times {(\frac{25}{9})}^{ \frac{ - 3}{2}}$

$\rule{200}{2}$

We know from the rules of Exponents and Radicals that :-

${a}^{ \frac{m}{n}} = \sqrt[n]{{a}^{m}}$

$\rule{200}{2}$

Applying this in the question we get :-

$\sqrt[4]{{(\frac{81}{16})}^{ - 3}} \times \sqrt{ {( \frac{25}{9})}^{ - 3} }$

$\rule{200}{2}$

Also we know that :-

${a}^{ - m} = \frac{1}{{a}^{m}}$

$\rule{200}{2}$

Applying this in the question we get again :-

$\sqrt[4]{{(\frac{81}{16})}^{ - 3}} \times \sqrt{ {( \frac{25}{9})}^{ - 3}} \\ \\ \\ = \sqrt[4]{{(\frac{16}{81})}^{3}} \times \sqrt{ {( \frac{9}{25})}^{3}} \\ \\ \\ = \sqrt[4]{ \frac{16 \times 16 \times 16}{81 \times 81 \times 81} } \times \sqrt{ {( \frac{9}{25})}^{ 3}} \\ \\ \\ = \frac{8}{27} \times \frac{9}{25} \times \frac{3}{5} \\ \\ \\ = \frac{8}{125}$

$\rule{200}{2}$

We know that :-

$16 = {2}^{4} \\ \\81 = {3}^{4} \\ \\ \\9 = {3}^{2} \\ \\25 = {5}^{2}$

Answer 2

Step-by-step explanation:

$Hey friend, \\ \\ </p><p>Here \: \: is \: \: the \: \: answer \: \: you \: were \: looking \: \: for: \\ \\ </p><p>\begin{gathered} {( \frac{81}{16} )}^{ - \frac{3}{4} } \times {( \frac{25}{9} )}^{ - \frac{3}{2} } \\ \\ = {( \frac{ {3}^{4} }{ {2}^{4} } })^{ - \frac{3}{4} } \times {( \frac{ {5}^{2} }{ {3}^{2} }) }^{ - \frac{3}{2} } \\ \\ = {( \frac{3}{2}) }^{4 \times - \frac{3}{4} } \times {( \frac{5}{3} )}^{2 \times - \frac{3}{2} } \\ \\ = {( \frac{3}{2} )}^{ - 3} \times {( \frac{5}{3}) }^{ - 3} \\ \\ = {( \frac{2}{3}) }^{3} \times {( \frac{3}{5} )}^{3} \\ \\ = \frac{2 \times 2 \times 2}{3 \times 3 \times 3} \times \frac{3 \times 3 \times 3}{5 \times 5 \times 5} \\ \\ = \frac{2 \times 2 \times 2}{5 \times 5 \times 5} \\ \\ = \frac{8}{125} \end{gathered} \\ \\ </p><p>Hope \: \: this \: \: helps!!!$