Solve for r cube root of r+21=8

$\sqrt[3]{r + 21} = 8$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{r + 21}}^{3} = 8^{3}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{r + 21}$ as ${(r + 21)}^{\frac{1}{3}}$ .

${({(r + 21)}^{\frac{1}{3}})}^{3} = 8^{3}$

Simplify the left side.

Simplify ${({(r + 21)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(r + 21)}^{\frac{1}{3}})}^{3}$ .

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

${(r + 21)}^{\frac{1}{3} \cdot 3} = 8^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(r + 21)}^{\frac{1}{3} \cdot 3} = 8^{3}$

Rewrite the expression.

${(r + 21)}^{1} = 8^{3}$

Simplify.

$r + 21 = 8^{3}$

Simplify the right side.

Raise $8$ to the power of $3$ .

$r + 21 = 512$

Step 3

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

Subtract $21$ from both sides of the equation.

$r = 512 - 21$

Subtract $21$ from $512$ .

$r = 491$