Solve for n (3^3)^2=n^6

${(3^{3})}^{2} = n^{6}$

Step 1

Rewrite the equation as $n^{6} = {(3^{3})}^{2}$ .

$n^{6} = {(3^{3})}^{2}$

Step 2

Take the 6th root of both sides of the equation to eliminate the exponent on the left side.

$n = \pm \sqrt[6]{{(3^{3})}^{2}}$

Step 3

Simplify $\pm \sqrt[6]{{(3^{3})}^{2}}$ .

Multiply the exponents in ${(3^{3})}^{2}$ .

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

$n = \pm \sqrt[6]{3^{3 \cdot 2}}$

Multiply $3$ by $2$ .

$n = \pm \sqrt[6]{3^{6}}$

Raise $3$ to the power of $6$ .

$n = \pm \sqrt[6]{729}$

Rewrite $729$ as $3^{6}$ .

$n = \pm \sqrt[6]{3^{6}}$

Pull terms out from under the radical, assuming positive real numbers.

$n = \pm 3$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$n = 3$

Next, use the negative value of the $\pm$ to find the second solution.

$n = - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$n = 3, - 3$