Solve for r r^(4/3)=81

$r^{\frac{4}{3}} = 81$

Step 1

Raise each side of the equation to the power of $\frac{3}{4}$ to eliminate the fractional exponent on the left side.

${(r^{\frac{4}{3}})}^{\frac{3}{4}} = \pm 81^{\frac{3}{4}}$

Step 2

Simplify the exponent.

Simplify the left side.

Simplify ${(r^{\frac{4}{3}})}^{\frac{3}{4}}$ .

Multiply the exponents in ${(r^{\frac{4}{3}})}^{\frac{3}{4}}$ .

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

$r^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 81^{\frac{3}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$r^{\frac{4}{3} \cdot \frac{3}{4}} = \pm 81^{\frac{3}{4}}$

Rewrite the expression.

$r^{\frac{1}{3} \cdot 3} = \pm 81^{\frac{3}{4}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$r^{\frac{1}{3} \cdot 3} = \pm 81^{\frac{3}{4}}$

Rewrite the expression.

$r^{1} = \pm 81^{\frac{3}{4}}$

Simplify.

$r = \pm 81^{\frac{3}{4}}$

Simplify the right side.

Simplify $\pm 81^{\frac{3}{4}}$ .

Simplify the expression.

Rewrite $81$ as $3^{4}$ .

$r = \pm {(3^{4})}^{\frac{3}{4}}$

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

$r = \pm 3^{4 (\frac{3}{4})}$

Cancel the common factor of $4$ .

Cancel the common factor.

$r = \pm 3^{4 (\frac{3}{4})}$

Rewrite the expression.

$r = \pm 3^{3}$

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

$r = \pm 27$

Step 3

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.

$r = 27$

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

$r = - 27$

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

$r = 27, - 27$