Question
Determine $(8\text{x})^\text{x},$ if $9^{\text{x}+2}=240+9^\text{x}. $
✓
Answer
$9^{\text{x}+2}=240+9^\text{x}$$\Rightarrow9^\text{x}\times9^2=240+9^\text{x}$
$\Rightarrow9^\text{x}\times81=240+9^\text{x}$
$\Rightarrow9^\text{x}\times81-9^\text{x}=240$
$\Rightarrow9^\text{x}(81-1)=240$
$\Rightarrow9^\text{x}\times80=240$
$\Rightarrow9^\text{x}=3$
$\Rightarrow\big(3^2\big)^\text{x}=3$
$\Rightarrow3^{2\text{x}}=3$
$\Rightarrow2\text{x}=1$
$\Rightarrow\text{x}=\frac{1}{2}$
$\Rightarrow(8\text{x})^\text{x}=\Big(8\times\frac{1}{2}\Big)^{\frac{1}{2}}=(4)^\frac{1}{2}=2^{2\times\frac{1}{2}}=2$
$\Rightarrow9^\text{x}\times81=240+9^\text{x}$
$\Rightarrow9^\text{x}\times81-9^\text{x}=240$
$\Rightarrow9^\text{x}(81-1)=240$
$\Rightarrow9^\text{x}\times80=240$
$\Rightarrow9^\text{x}=3$
$\Rightarrow\big(3^2\big)^\text{x}=3$
$\Rightarrow3^{2\text{x}}=3$
$\Rightarrow2\text{x}=1$
$\Rightarrow\text{x}=\frac{1}{2}$
$\Rightarrow(8\text{x})^\text{x}=\Big(8\times\frac{1}{2}\Big)^{\frac{1}{2}}=(4)^\frac{1}{2}=2^{2\times\frac{1}{2}}=2$
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