Solve for z 0 = cube root of 2(462)+z-12

$0 = \sqrt[3]{2 (462) + z} - 12$

Step 1

Rewrite the equation as $\sqrt[3]{2 (462) + z} - 12 = 0$ .

$\sqrt[3]{2 (462) + z} - 12 = 0$

Step 2

Add $12$ to both sides of the equation.

$\sqrt[3]{2 (462) + z} = 12$

Step 3

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

${\sqrt[3]{2 (462) + z}}^{3} = 12^{3}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 (462) + z}$ as ${(2 (462) + z)}^{\frac{1}{3}}$ .

${({(2 (462) + z)}^{\frac{1}{3}})}^{3} = 12^{3}$

Simplify the left side.

Simplify ${({(2 (462) + z)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(2 (462) + z)}^{\frac{1}{3}})}^{3}$ .

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

${(2 (462) + z)}^{\frac{1}{3} \cdot 3} = 12^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(2 (462) + z)}^{\frac{1}{3} \cdot 3} = 12^{3}$

Rewrite the expression.

${(2 (462) + z)}^{1} = 12^{3}$

Multiply $2$ by $462$ .

${(924 + z)}^{1} = 12^{3}$

Simplify.

$924 + z = 12^{3}$

Simplify the right side.

Raise $12$ to the power of $3$ .

$924 + z = 1728$

Step 5

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

Subtract $924$ from both sides of the equation.

$z = 1728 - 924$

Subtract $924$ from $1728$ .

$z = 804$