Question

(81/16)to the power of -3/4 × (25/9) to the power of -3/2

Answer 1

$$\begin{gathered} {\left(\frac{81}{16}\right)}^{-3/4}\times {\left(\frac{25}{9}\right)}^{-3/2} \\ ={\left[{\left(\frac{3}{2}\right)}^4\right]}^{-3/4}\times {\left[{\left(\frac{5}{3}\right)}^2\right]}^{-3/2} \\ ={\left(\frac{3}{2}\right)}^{4\times -\frac{3}{4}}\times {\left(\frac{5}{3}\right)}^{2\times -\frac{3}{2}}\left[\mathbf{as}\mathbf{,}\mathbf{}{\left({\mathbf{a}}^{\mathbf{m}}\right)}^{\mathbf{n}}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{a}}^{\mathbf{mn}}\right] \\ ={\left(\frac{3}{2}\right)}^{-3}\times {\left(\frac{5}{3}\right)}^{-3} \\ ={\left(\frac{2}{3}\right)}^3\times {\left(\frac{3}{5}\right)}^3\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[\mathbf{as}\mathbf{,}\mathbf{}{\mathbf{a}}^{\mathbf{-}\mathbf{m}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{{\mathbf{a}}^{\mathbf{m}}}\right] \\ ={\left(\frac{2}{3}\times \frac{3}{5}\right)}^3\mathbf{}\left[\mathbf{as}\mathbf{,}\mathbf{}{\mathbf{a}}^{\mathbf{m}}\mathbf{\times }{\mathbf{b}}^{\mathbf{m}}\mathbf{}\mathbf{=}\mathbf{}{\left(\mathbf{a}\mathbf{\times }\mathbf{b}\right)}^{\mathbf{m}}\right] \\ ={\left(\frac{2}{5}\right)}^3 \\ \mathbf{=}\frac{\mathbf{8}}{\mathbf{125}} \end{gathered}$$