Question

No decimals. Answer must be in integers and reduced fractions

Answer 1

The equation is given to be:

$36^{n-3}\cdot\: 216^n=216^{2n+1}$

STEP 1: Rewrite each term in terms of 6.

We have that:

$\begin{gathered} 36=6^2 \\ 216=6^3 \end{gathered}$

Therefore, the equation becomes:

$\Rightarrow6^{2(n-3)}\cdot6^{3n}=6^{3(2n+1)}$

STEP 2: Apply the exponent rule:

$a^m\times a^n=a^{m+n}$

Therefore, we can have the equation to be:

$6^{\lbrack2(n-3)+3n\rbrack}=6^{3(2n+1)}$

STEP 3: Compare the exponents, since the bases are the same.

$2(n-3)+3n=3(2n+1)$

STEP 4: Expand the parentheses using the Distributive Property.

$\begin{gathered} 2n-6+3n=6n+3 \\ 5n-6=6n+3 \end{gathered}$

STEP 5: Subtract 5n and 3 from both sides.

$\begin{gathered} 5n-6-5n-3=6n+3-5n-3 \\ -9=n \end{gathered}$

ANSWER: The answer is:

$n=-9$