Solve for n 27^(3n)=81^(2n+1)

$27^{3 n} = 81^{2 n + 1}$

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

$3^{3 (3 n)} = 3^{4 (2 n + 1)}$

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

$3 (3 n) = 4 (2 n + 1)$

Solve for $n$ .

Multiply $3$ by $3$ .

$9 n = 4 (2 n + 1)$

Simplify $4 (2 n + 1)$ .

Apply the distributive property.

$9 n = 4 (2 n) + 4 \cdot 1$

Multiply.

Multiply $2$ by $4$ .

$9 n = 8 n + 4 \cdot 1$

Multiply $4$ by $1$ .

$9 n = 8 n + 4$

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

Subtract $8 n$ from both sides of the equation.

$9 n - 8 n = 4$

Subtract $8 n$ from $9 n$ .

$n = 4$