Question

Evaluate:
$\left(viii\right)$ $9^0+9^{-1}-9^{-2}+9^{\frac{1}{2}}-9^{\frac{-1}{2}}$

Answer 1

Simplify the given expression

Given,$9^0+9^{-1}-9^{-2}+9^{\frac{1}{2}}-9^{\frac{-1}{2}}$

We know that $a^{-m}=\frac{1}{a^m};a^0=1,$
$\Rightarrow 9^0+9^{-1}-9^{-2}+9^{\frac{1}{2}}-9^{\frac{-1}{2}}$
$=1+\frac{1}{9}-\frac{1}{9^2}+9^{\frac{1}{2}}-\frac{1}{9^{\frac{1}{2}}}$
$=1+\frac{1}{9}-\frac{1}{81}+3^{2\times \frac{1}{2}}-\frac{1}{3^{2\times \frac{1}{2}}}$
$=1+\frac{1}{9}-\frac{1}{81}+3-\frac{1}{3}$

$=\frac{81+9-1+243-27}{81}$
$=\frac{333-28}{81}=\frac{305}{81}=3\frac{62}{81}$
Hence, $9^0+9^{-1}-9^{-2}+9^{\frac{1}{2}}-9^{\frac{-1}{2}}$
$=\frac{305}{81}or3\frac{62}{81}$