Solve for r r^(4/7)=16

$r^{\frac{4}{7}} = 16$

Step 1

Raise each side of the equation to the power of $\frac{7}{4}$ to eliminate the fractional exponent on the left side.

${(r^{\frac{4}{7}})}^{\frac{7}{4}} = \pm 16^{\frac{7}{4}}$

Step 2

Simplify the exponent.

Simplify the left side.

Simplify ${(r^{\frac{4}{7}})}^{\frac{7}{4}}$ .

Multiply the exponents in ${(r^{\frac{4}{7}})}^{\frac{7}{4}}$ .

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

$r^{\frac{4}{7} \cdot \frac{7}{4}} = \pm 16^{\frac{7}{4}}$

Cancel the common factor of $4$ .

Cancel the common factor.

$r^{\frac{4}{7} \cdot \frac{7}{4}} = \pm 16^{\frac{7}{4}}$

Rewrite the expression.

$r^{\frac{1}{7} \cdot 7} = \pm 16^{\frac{7}{4}}$

Cancel the common factor of $7$ .

Cancel the common factor.

$r^{\frac{1}{7} \cdot 7} = \pm 16^{\frac{7}{4}}$

Rewrite the expression.

$r^{1} = \pm 16^{\frac{7}{4}}$

Simplify.

$r = \pm 16^{\frac{7}{4}}$

Simplify the right side.

Simplify $\pm 16^{\frac{7}{4}}$ .

Simplify the expression.

Rewrite $16$ as $2^{4}$ .

$r = \pm {(2^{4})}^{\frac{7}{4}}$

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

$r = \pm 2^{4 (\frac{7}{4})}$

Cancel the common factor of $4$ .

Cancel the common factor.

$r = \pm 2^{4 (\frac{7}{4})}$

Rewrite the expression.

$r = \pm 2^{7}$

Raise $2$ to the power of $7$ .

$r = \pm 128$

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.

$r = 128$

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

$r = - 128$

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

$r = 128, - 128$