Simplify \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}}\)
Question 1 Report
Simplify \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}}\)
Answer Details
We can simplify the given expression using the rules of exponents and radicals as follows:
\begin{align*}
\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}} &= \left(\frac{81}{16}\right)^{\frac{3}{4}}\times \frac{10}{9} \\
&= \left[\left(\frac{3^4}{2^4}\right)^{\frac{1}{4}}\right]^3 \times \frac{10}{9} \\
&= \left(\frac{3}{2}\right)^3 \times \frac{10}{9} \\
&= \frac{27 \times 10}{8 \times 9} \\
&= \frac{15}{4}
\end{align*}
Therefore, the simplified expression is \(\frac{15}{4}\). So, the answer is option (D).
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