Solve for x fifth root of x+7=-2

$\sqrt[5]{x + 7} = - 2$

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{x + 7}}^{5} = {(- 2)}^{5}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x + 7}$ as ${(x + 7)}^{\frac{1}{5}}$ .

${({(x + 7)}^{\frac{1}{5}})}^{5} = {(- 2)}^{5}$

Simplify the left side.

Simplify ${({(x + 7)}^{\frac{1}{5}})}^{5}$ .

Multiply the exponents in ${({(x + 7)}^{\frac{1}{5}})}^{5}$ .

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

${(x + 7)}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x + 7)}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Rewrite the expression.

${(x + 7)}^{1} = {(- 2)}^{5}$

Simplify.

$x + 7 = {(- 2)}^{5}$

Simplify the right side.

Raise $- 2$ to the power of $5$ .

$x + 7 = - 32$

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

Subtract $7$ from both sides of the equation.

$x = - 32 - 7$

Subtract $7$ from $- 32$ .

$x = - 39$