Correct option : (b) \(\frac92\)
\(\sqrt[3]{\left[\left\{(128)^{-5}\right\}^{3/7}\right]^{-1/5}}\) \(=\left((2^7)^{-5\times \frac37\times\frac{-1}{5}}\right) =(2^3)^{\frac13} =2\)
\(\left[(8^{-3/4})^{5/2}\right]^{8/15} =(2^3)^{\frac{-3}{4}\times\frac{5}{2}\times \frac{8}{15}} =(2^3)^{-1} =2^{-3}=\frac18.\)
\(\sqrt{\frac{(12.12)^2-(8.12)^2}{(0.25)^2+(0.25)(19.99)}}=\)\(\sqrt{\frac{(12.12 + 8.12)(12.12-8.12)}{(0.25)(0.25+19.99)}}\)
\(=\sqrt{\frac{4\times20.24}{0.25\times20.24}} =\sqrt{\frac{2^2}{(0.5)^2}} =\frac{2}{0.5}=\cfrac{2}{\frac12}=4.\)
\(\therefore\) Given Exp.\(=\cfrac{4 + \frac18 \times(2^4)^\frac34}{2} =\cfrac{4+\frac18 \times8}{2}\)
\(=4+\frac12 =\frac92.\)
