Solve for n (x^2)^n*x^3=x^9

${(x^{2})}^{n} \cdot x^{3} = x^{9}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{2 n} \cdot x^{3} = x^{9}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 n + 3} = x^{9}$

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

$2 n + 3 = 9$

Solve for $n$ .

Move all terms not containing $n$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$2 n = 9 - 3$

Subtract $3$ from $9$ .

$2 n = 6$

Divide each term in $2 n = 6$ by $2$ and simplify.

Divide each term in $2 n = 6$ by $2$ .

$\frac{2 n}{2} = \frac{6}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 n}{2} = \frac{6}{2}$

Divide $n$ by $1$ .

$n = \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$n = 3$