Correct Answer: Option D

Explanation

\(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}}\)
= \(\frac{1}{\left(\sqrt[4]{\frac{16}{81}}\right)^3}\times \frac{10}{9}\)

= \(\frac{1}{\left(\frac{2}{3}\right)^3}\times\frac{10}{9}\)
= \(\frac{27}{8}\times \frac{10}{9} = \frac{15}{4}\)                      OR

We start with the expression:
\(\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times \sqrt{\frac{100}{81}}\)

First, simplify \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\):
\(\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \left(\frac{81}{16}\right)^{\frac{3}{4}} = \frac{81^{\frac{3}{4}}}{16^{\frac{3}{4}}}\)
Calculating \(81^{\frac{3}{4}}\) and \(16^{\frac{3}{4}}\):
\(81 = 3^4 \implies 81^{\frac{3}{4}} = 3^3\) = 27
\(16 = 4^2 \implies 16^{\frac{3}{4}} = 4^{\frac{3}{2}} = 2^3\) = 8
Thus,
\(\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \frac{27}{8}\)

Next, simplify \(\sqrt{\frac{100}{81}}\):
\(\sqrt{\frac{100}{81}} = \frac{\sqrt{100}}{\sqrt{81}} = \frac{10}{9}\)

Now, multiply the two results:
\(\frac{27}{8} \times \frac{10}{9} = \frac{27 \times 10}{8 \times 9} = \frac{270}{72}\)

Finding the GCD of 270 and 72 gives:
\(\frac{270 \div 18}{72 \div 18} = \frac{15}{4}\)

The final result is:
\(\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times \sqrt{\frac{100}{81}} = \frac{15}{4}\)

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