Answer
The solution set is $\left\{6\right\}$.
Work Step by Step
To solve the given equation, make the two sides have the same base.
Note that $9 = 3^2$ and $27=3^3$, so the given equation is equivalent to:
$(3^2)^{-x+15}=(3^3)^x$
Use the rule $(a^m)^n = a^{mn}$ to obtain:
$3^{2(-x+15)} = 3^{3x}
\\3^{-2x+30}=3^{3x}$
Use the rule $a^m=a^n \longrightarrow m=n$ to obtain:
$-2x+30=3x$
Add $2x$ to both sides of the equation to obtain:
$\begin{array}{ccc}
&-2x+30+2x &= &3x+2x
\\&30 &= &5x
\end{array}$
Divide by 5 on both sides of the equation to obtain:
$\begin{array}{ccc}
&\frac{30}{5} &= &\frac{5x}{5}
\\&6 &= &x
\end{array}$
Thus, the solution set is $\left\{6\right\}$.
