The equation is given to be:
[tex]36^{n-3}\cdot\: 216^n=216^{2n+1}[/tex]
STEP 1: Rewrite each term in terms of 6.
We have that:
[tex]\begin{gathered} 36=6^2 \\ 216=6^3 \end{gathered}[/tex]
Therefore, the equation becomes:
[tex]\Rightarrow6^{2(n-3)}\cdot6^{3n}=6^{3(2n+1)}[/tex]
STEP 2: Apply the exponent rule:
[tex]a^m\times a^n=a^{m+n}[/tex]
Therefore, we can have the equation to be:
[tex]6^{\lbrack2(n-3)+3n\rbrack}=6^{3(2n+1)}[/tex]
STEP 3: Compare the exponents, since the bases are the same.
[tex]2(n-3)+3n=3(2n+1)[/tex]
STEP 4: Expand the parentheses using the Distributive Property.
[tex]\begin{gathered} 2n-6+3n=6n+3 \\ 5n-6=6n+3 \end{gathered}[/tex]
STEP 5: Subtract 5n and 3 from both sides.
[tex]\begin{gathered} 5n-6-5n-3=6n+3-5n-3 \\ -9=n \end{gathered}[/tex]
ANSWER: The answer is:
[tex]n=-9[/tex]
