Solve for a b=1/2 cube root of a-1

$b = \frac{1}{2} \sqrt[3]{a - 1}$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot \sqrt[3]{a - 1} = b$ .

$\frac{1}{2} \cdot \sqrt[3]{a - 1} = b$

Step 2

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot \sqrt[3]{a - 1}) = 2 b$

Step 3

Simplify the left side.

Simplify $2 (\frac{1}{2} \cdot \sqrt[3]{a - 1})$ .

Combine $\frac{1}{2}$ and $\sqrt[3]{a - 1}$ .

$2 \frac{\sqrt[3]{a - 1}}{2} = 2 b$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{\sqrt[3]{a - 1}}{2} = 2 b$

Rewrite the expression.

$\sqrt[3]{a - 1} = 2 b$

Step 4

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

${\sqrt[3]{a - 1}}^{3} = {(2 b)}^{3}$

Step 5

Simplify each side of the equation.

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

${({(a - 1)}^{\frac{1}{3}})}^{3} = {(2 b)}^{3}$

Simplify the left side.

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

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

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

${(a - 1)}^{\frac{1}{3} \cdot 3} = {(2 b)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(a - 1)}^{\frac{1}{3} \cdot 3} = {(2 b)}^{3}$

Rewrite the expression.

${(a - 1)}^{1} = {(2 b)}^{3}$

Simplify.

$a - 1 = {(2 b)}^{3}$

Simplify the right side.

Simplify ${(2 b)}^{3}$ .

Apply the product rule to $2 b$ .

$a - 1 = 2^{3} b^{3}$

Raise $2$ to the power of $3$ .

$a - 1 = 8 b^{3}$

Step 6

Add $1$ to both sides of the equation.

$a = 8 b^{3} + 1$