Q. Simplify
(8³  x 2^-7 ) x (4^-5 x 7³ ) / 21^-4 x (8⁵) x (81)^​-3

Dear Student,

Please find below the solution to the asked query:

We have : $\frac{\left(8^3 \times 2^{-7}\right) \times \left(4^{-5} \times 7^{3 }\right)}{{21}^{-4} \times \left(8^5\right)\times {\left(81\right)}^{-3}}$ , So

$$\begin{gathered} \Rightarrow \frac{\left({\left(2^3\right)}^3 \times 2^{-7}\right) \times \left({\left(2^2\right)}^{-5} \times 7^{3 }\right)}{{\left( 3 \times 7\right)}^{-4} \times \left({\left(2^3\right)}^5\right)\times {\left(3^4\right)}^{-3}} \\ \Rightarrow \frac{\left(2^9\times 2^{-7}\right) \times \left(2^{- 10}\times 7^{3 }\right)}{{\left( 3 \times 7\right)}^{-4} \times 2^{15}\times 3^{- 12}} ( \mathrm{We} \mathrm{know} {\left({\mathrm{a}}^{\mathrm{m}}\right)}^{\mathrm{n}} = {\mathrm{a}}^{\mathrm{m} \times \mathrm{n}} ) \\ \Rightarrow \frac{ 2^9\times 2^{-7} \times 2^{- 10}\times 7^{3 }}{3^{-4} \times 7^{-4}\times 2^{15}\times 3^{- 12}} ( \mathrm{We} \mathrm{know} {\left(\mathrm{a} \times \mathrm{b}\right)}^{\mathrm{m}} = {\mathrm{a}}^{\mathrm{m} }\times {\mathrm{b}}^{\mathrm{m}} ) \\ \Rightarrow \frac{2^{- 8} \times 7^{3 }}{3^{-16} \times 7^{-4}\times 2^{15} } ( \mathrm{We} \mathrm{know} {\mathrm{a}}^{\mathrm{m}} \times {\mathrm{a}}^{\mathrm{n}} = {\mathrm{a}}^{\mathrm{m} + \mathrm{n}} ) \\ \Rightarrow 2^{- 8- 15} \times 7^{3- \left(- 4 \right) }\times 3^{16} ( \mathrm{We} \mathrm{know} \frac{{\mathrm{a}}^{\mathrm{m}} }{{\mathrm{a}}^{\mathrm{n}} }= {\mathrm{a}}^{\mathrm{m} - \mathrm{n}} \mathrm{and} {\mathrm{a}}^{\mathrm{m}} = \frac{1}{{\mathrm{a}}^{-\mathrm{m}}}) \\ \mathbf{\Rightarrow }{\mathbf{2}}^{\mathbf{-}\mathbf{ }\mathbf{23}}\mathbf{ }\mathbf{ }\mathbf{\times }\mathbf{ }{\mathbf{7}}^{\mathbf{7}}\mathbf{\times }{\mathbf{3}}^{\mathbf{16}}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{(}\mathbf{ }\mathbf{Ans}\mathbf{ }\mathbf{)} \end{gathered}$$

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