Solve for x y = cube root of x-2

$y = \sqrt[3]{x - 2}$

Rewrite the equation as $\sqrt[3]{x - 2} = y$ .

$\sqrt[3]{x - 2} = y$

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

${\sqrt[3]{x - 2}}^{3} = y^{3}$

Simplify each side of the equation.

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

${({(x - 2)}^{\frac{1}{3}})}^{3} = y^{3}$

Simplify the left side.

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

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

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

${(x - 2)}^{\frac{1}{3} \cdot 3} = y^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{3} \cdot 3} = y^{3}$

Rewrite the expression.

${(x - 2)}^{1} = y^{3}$

Simplify.

$x - 2 = y^{3}$

Add $2$ to both sides of the equation.

$x = y^{3} + 2$