Question

Simplify:
$\left(iii\right)$ $(2^{-3}-2^{-4})(2^{-3}+2^{-4})$

Answer 1

Simplify the given expression

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

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

$[\therefore a^{-m}=\frac{1}{a^m}]$

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

$[\therefore (a+b\left)\right(a-b)=a^2-b^2]$

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

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

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