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* 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`