Q. Simplify [ ( 7  ÷ 8)^-1 -  ( 7  ÷ 6)^-1] ^-2 ÷  ( 6 - 8 )^-6

Dear Student,

Please find below the solution to the asked query:

We have : ${\left[{\left(7 ÷ 8 \right)}^{-1} - {\left(7 ÷ 6 \right)}^{-1}\right]}^{-2}÷{\left( 6 - 8 \right)}^{-6}$ , So

$$\begin{gathered} \Rightarrow {\left[\left(8 ÷7 \right)- \left(6 ÷ 7 \right)\right]}^{-2}÷{\left( 6 - 8 \right)}^{-6} ( \mathrm{We} \mathrm{know} \frac{1}{{\mathrm{a}}^{\mathrm{m}}} = {\mathrm{a}}^{-\mathrm{m}} ) \\ \Rightarrow {\left[\frac{8}{7}- \frac{6}{7}\right]}^{-2}÷{\left(- 2 \right)}^{-6} \\ \Rightarrow {\left[\frac{8 - 6}{7}\right]}^{-2}÷\frac{1}{{\left(- 2 \right)}^6} \\ \Rightarrow {\left[\frac{2}{7}\right]}^{-2}÷\frac{1}{2^6} \\ \Rightarrow {\left[\frac{7}{2}\right]}^2÷{\left(\frac{1}{2}\right)}^6 \\ \Rightarrow \frac{7^2}{2^2}\times 2^6 \\ \Rightarrow 7^2\times 2^4 ( \mathrm{We} \mathrm{know} \frac{{\mathrm{a}}^{\mathrm{m}}}{{\mathrm{a}}^{\mathrm{n}}} = {\mathrm{a}}^{\mathrm{m} - \mathrm{n}} ) \\ \Rightarrow 49\times 16 \\ \mathbf{\Rightarrow }\mathbf{784}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$

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