Question
If $3^{\text{x}+1}=9^{\text{x}-2},$ find the value of $2^{1+\text{x}}.$
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Answer
$3^{\text{x}+1}=9^{\text{x}-2}$$\Rightarrow3^\text{x}\times3=9^\text{x}\times9^{-2}$
$\Rightarrow3^\text{x}\times3=(3^2)^\text{x}\times\frac{1}{9^2}$
$\Rightarrow3^\text{x}\times3=3^{2\text{x}}\times\frac{1}{(3^2)^2}$
$\Rightarrow3^\text{x}\times3\times3^{2\text{x}}\times\frac{1}{3^4}$
$\Rightarrow\frac{3^{2\text{x}}}{3^\text{x}}=3\times3^4$
$\Rightarrow3^{2\text{x}-\text{x}}=3^5$
$\Rightarrow3^\text{x}=3^5$
$\Rightarrow\text{x}=5$
$\Rightarrow2^{1+\text{x}}=2^{1+5}=2^6=64$
$\Rightarrow3^\text{x}\times3=(3^2)^\text{x}\times\frac{1}{9^2}$
$\Rightarrow3^\text{x}\times3=3^{2\text{x}}\times\frac{1}{(3^2)^2}$
$\Rightarrow3^\text{x}\times3\times3^{2\text{x}}\times\frac{1}{3^4}$
$\Rightarrow\frac{3^{2\text{x}}}{3^\text{x}}=3\times3^4$
$\Rightarrow3^{2\text{x}-\text{x}}=3^5$
$\Rightarrow3^\text{x}=3^5$
$\Rightarrow\text{x}=5$
$\Rightarrow2^{1+\text{x}}=2^{1+5}=2^6=64$
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