Question

Determine $(8x{)}^x,$ if $9^{x+2}=240+9^x$.

Answer 1

Given : $9^{x+2}=240+9^x$

$\Rightarrow 9^x\times 9^2=240+9^x$

$\Rightarrow 9^x\times 81-9^x=240$

$\Rightarrow 9^x(81-1)=240$

$\Rightarrow 9^x=\frac{240}{80}$

$\Rightarrow (3^2{)}^x=3$

$\Rightarrow 3^{2x}=3^1$

On comparing both the sides, we get

$\Rightarrow 2x=1$

$\Rightarrow x=\frac{1}{2}$

$\therefore (8x{)}^x={(8\times \frac{1}{2})}^{\frac{1}{2}}$

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

$=(2^2{)}^{\frac{1}{2}}$

$=2$

Hence, the required value is $2$.