Solve for n 64^(n+2)=16^(3n)

$64^{n + 2} = 16^{3 n}$

Create equivalent expressions in the equation that all have equal bases.

$4^{3 (n + 2)} = 4^{2 (3 n)}$

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

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

Solve for $n$ .

Simplify $3 (n + 2)$ .

Start simplifying.

$0 + 0 + 3 (n + 2) = 2 \cdot (3 n)$

Simplify by adding zeros.

$3 (n + 2) = 2 \cdot (3 n)$

Apply the distributive property.

$3 n + 3 \cdot 2 = 2 \cdot (3 n)$

Multiply $3$ by $2$ .

$3 n + 6 = 2 \cdot (3 n)$

Multiply $3$ by $2$ .

$3 n + 6 = 6 n$

Move all terms containing $n$ to the left side of the equation.

Subtract $6 n$ from both sides of the equation.

$3 n + 6 - 6 n = 0$

Subtract $6 n$ from $3 n$ .

$- 3 n + 6 = 0$

Subtract $6$ from both sides of the equation.

$- 3 n = - 6$

Divide each term in $- 3 n = - 6$ by $- 3$ and simplify.

Divide each term in $- 3 n = - 6$ by $- 3$ .

$\frac{- 3 n}{- 3} = \frac{- 6}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 n}{- 3} = \frac{- 6}{- 3}$

Divide $n$ by $1$ .

$n = \frac{- 6}{- 3}$

Simplify the right side.

Divide $- 6$ by $- 3$ .

$n = 2$