Simplify:
$(i i i)$ $(2^{- 3} - 2^{- 4}) (2^{- 3} + 2^{- 4})$

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Solution

Simplify the given expression

Given, $(2^{- 3} - 2^{- 4}) (2^{- 3} + 2^{- 4})$

$= (\frac{1}{2^{3}} - \frac{1}{2^{4}}) (\frac{1}{2^{3}} + \frac{1}{2^{4}})$

$[∴ a^{- m} = \frac{1}{a^{m}}]$

$= {(\frac{1}{2^{3}})}^{2}$ - ${(\frac{1}{2^{4}})}^{2}$

$[∴ (a + b) (a - b) = a^{2} - b^{2}]$

$= {(\frac{1}{8})}^{2}$ $-$ ${(\frac{1}{16})}^{2}$

$= \frac{1}{64} - \frac{1}{256} - \frac{4 - 1}{256} = \frac{3}{256}$

Hence, $(2^{- 3} - 2^{- 4}) (2^{- 3} + 2^{- 4}) = \frac{3}{256}$

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