Solve for x cube root of 2x+1 = cube root of x+8

$\sqrt[3]{2 x + 1} = \sqrt[3]{x + 8}$

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

${\sqrt[3]{2 x + 1}}^{3} = {\sqrt[3]{x + 8}}^{3}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x + 1}$ as ${(2 x + 1)}^{\frac{1}{3}}$ .

${({(2 x + 1)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{x + 8}}^{3}$

Simplify the left side.

Simplify ${({(2 x + 1)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(2 x + 1)}^{\frac{1}{3}})}^{3}$ .

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

${(2 x + 1)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{x + 8}}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(2 x + 1)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{x + 8}}^{3}$

Rewrite the expression.

${(2 x + 1)}^{1} = {\sqrt[3]{x + 8}}^{3}$

Simplify.

$2 x + 1 = {\sqrt[3]{x + 8}}^{3}$

Simplify the right side.

Rewrite ${\sqrt[3]{x + 8}}^{3}$ as $x + 8$ .

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

$2 x + 1 = {({(x + 8)}^{\frac{1}{3}})}^{3}$

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

$2 x + 1 = {(x + 8)}^{\frac{1}{3} \cdot 3}$

Combine $\frac{1}{3}$ and $3$ .

$2 x + 1 = {(x + 8)}^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$2 x + 1 = {(x + 8)}^{\frac{3}{3}}$

Rewrite the expression.

$2 x + 1 = {(x + 8)}^{1}$

Simplify.

$2 x + 1 = x + 8$

Solve for $x$ .

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$2 x + 1 - x = 8$

Subtract $x$ from $2 x$ .

$x + 1 = 8$

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

Subtract $1$ from both sides of the equation.

$x = 8 - 1$

Subtract $1$ from $8$ .

$x = 7$