Precalculus Examples

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

Step 1

Multiply $\frac{1}{\sqrt[3]{a} - \sqrt[3]{b}}$ by $\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}$ .

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

Step 2

Combine fractions.

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Step 2.1

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{(\sqrt[3]{a} - \sqrt[3]{b}) ({\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{{\sqrt[3]{a}}^{3} + \sqrt[3]{a^{2} b} + \sqrt[3]{a} {(- \sqrt[3]{b})}^{2} - \sqrt[3]{b} {\sqrt[3]{a}}^{2} - \sqrt[3]{a b^{2}} - \sqrt[3]{b} {(- \sqrt[3]{b})}^{2}}$

Step 2.3

Simplify.

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Step 2.3.1

Subtract $\sqrt[3]{a^{2} b}$ from $\sqrt[3]{a^{2} b}$ .

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{a + 0 + \sqrt[3]{a b^{2}} - \sqrt[3]{a b^{2}} - b}$

Step 2.3.2

Add $a$ and $0$ .

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{a + \sqrt[3]{a b^{2}} - \sqrt[3]{a b^{2}} - b}$

Step 2.3.3

Subtract $\sqrt[3]{a b^{2}}$ from $\sqrt[3]{a b^{2}}$ .

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{a + 0 - b}$

Step 2.3.4

Add $a$ and $0$ .

$\frac{{\sqrt[3]{a}}^{2} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{a - b}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{a}}^{2}$ as $\sqrt[3]{a^{2}}$ .

$\frac{\sqrt[3]{a^{2}} - \sqrt[3]{a} (- \sqrt[3]{b}) + {(- \sqrt[3]{b})}^{2}}{a - b}$

Step 3.2

Multiply $- \sqrt[3]{a} (- \sqrt[3]{b})$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

Multiply $\sqrt[3]{a}$ by $1$ .

$\frac{\sqrt[3]{a^{2}} + \sqrt[3]{a} \sqrt[3]{b} + {(- \sqrt[3]{b})}^{2}}{a - b}$

Step 3.2.3

Combine using the product rule for radicals.

$\frac{\sqrt[3]{a^{2}} + \sqrt[3]{a b} + {(- \sqrt[3]{b})}^{2}}{a - b}$

Step 3.3

Apply the product rule to $- \sqrt[3]{b}$ .

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

Step 3.4

Raise $- 1$ to the power of $2$ .

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

Step 3.5

Multiply ${\sqrt[3]{b}}^{2}$ by $1$ .

$\frac{\sqrt[3]{a^{2}} + \sqrt[3]{a b} + {\sqrt[3]{b}}^{2}}{a - b}$

Step 3.6

Rewrite ${\sqrt[3]{b}}^{2}$ as $\sqrt[3]{b^{2}}$ .

$\frac{\sqrt[3]{a^{2}} + \sqrt[3]{a b} + \sqrt[3]{b^{2}}}{a - b}$