Find All Complex Solutions cube root of x+8=-4

$\sqrt[3]{x + 8} = - 4$

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

${\sqrt[3]{x + 8}}^{3} = {(- 4)}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 8}$ as ${(x + 8)}^{\frac{1}{3}}$ .

${({(x + 8)}^{\frac{1}{3}})}^{3} = {(- 4)}^{3}$

Simplify the left side.

Simplify ${({(x + 8)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 8)}^{\frac{1}{3}})}^{3}$ .

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

${(x + 8)}^{\frac{1}{3} \cdot 3} = {(- 4)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 8)}^{\frac{1}{3} \cdot 3} = {(- 4)}^{3}$

Rewrite the expression.

${(x + 8)}^{1} = {(- 4)}^{3}$

Simplify.

$x + 8 = {(- 4)}^{3}$

Simplify the right side.

Raise $- 4$ to the power of $3$ .

$x + 8 = - 64$

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

Subtract $8$ from both sides of the equation.

$x = - 64 - 8$

Subtract $8$ from $- 64$ .

$x = - 72$