Solve for a cube root of 432=a cube root of b

$\sqrt[3]{432} = a \sqrt[3]{b}$

Step 1

Rewrite the equation as $a \sqrt[3]{b} = \sqrt[3]{432}$ .

$a \sqrt[3]{b} = \sqrt[3]{432}$

Step 2

Divide each term in $a \sqrt[3]{b} = \sqrt[3]{432}$ by $\sqrt[3]{b}$ and simplify.

Divide each term in $a \sqrt[3]{b} = \sqrt[3]{432}$ by $\sqrt[3]{b}$ .

$\frac{a \sqrt[3]{b}}{\sqrt[3]{b}} = \frac{\sqrt[3]{432}}{\sqrt[3]{b}}$

Simplify the left side.

Cancel the common factor of $\sqrt[3]{b}$ .

Cancel the common factor.

$\frac{a \sqrt[3]{b}}{\sqrt[3]{b}} = \frac{\sqrt[3]{432}}{\sqrt[3]{b}}$

Divide $a$ by $1$ .

$a = \frac{\sqrt[3]{432}}{\sqrt[3]{b}}$

Simplify the right side.

Simplify the numerator.

Rewrite $432$ as $6^{3} \cdot 2$ .

Factor $216$ out of $432$ .

$a = \frac{\sqrt[3]{216 (2)}}{\sqrt[3]{b}}$

Rewrite $216$ as $6^{3}$ .

$a = \frac{\sqrt[3]{6^{3} \cdot 2}}{\sqrt[3]{b}}$

Pull terms out from under the radical.

$a = \frac{6 \sqrt[3]{2}}{\sqrt[3]{b}}$

Multiply $\frac{6 \sqrt[3]{2}}{\sqrt[3]{b}}$ by $\frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$ .

$a = \frac{6 \sqrt[3]{2}}{\sqrt[3]{b}} \cdot \frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$

Combine and simplify the denominator.

Multiply $\frac{6 \sqrt[3]{2}}{\sqrt[3]{b}}$ by $\frac{{\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{2}}$ .

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{\sqrt[3]{b} {\sqrt[3]{b}}^{2}}$

Raise $\sqrt[3]{b}$ to the power of $1$ .

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{1} {\sqrt[3]{b}}^{2}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{1 + 2}}$

Add $1$ and $2$ .

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{\sqrt[3]{b}}^{3}}$

Rewrite ${\sqrt[3]{b}}^{3}$ as $b$ .

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

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{{(b^{\frac{1}{3}})}^{3}}$

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

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{1}{3} \cdot 3}}$

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

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{3}{3}}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{\frac{3}{3}}}$

Rewrite the expression.

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b^{1}}$

Simplify.

$a = \frac{6 \sqrt[3]{2} {\sqrt[3]{b}}^{2}}{b}$

Simplify the numerator.

Rewrite ${\sqrt[3]{b}}^{2}$ as ${(b^{2})}^{\frac{1}{3}}$ .

$a = \frac{6 \sqrt[3]{2} \sqrt[3]{b^{2}}}{b}$

Combine using the product rule for radicals.

$a = \frac{6 \sqrt[3]{b^{2} \cdot 2}}{b}$

Move $2$ to the left of $b^{2}$ .

$a = \frac{6 \sqrt[3]{2 b^{2}}}{b}$

Solve for a cube root of 432=a cube root of b

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