Algebra Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{1^{2} - 1 (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2}}{- 1}$ .

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

Step 2.4.2

Rewrite $- 1 \cdot (1^{2} - 1 (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})$ as $- (1^{2} - 1 (- \sqrt[3]{2}) + {(- \sqrt[3]{2})}^{2})$ .

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

Step 3

Simplify each term.

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Step 3.1

One to any power is one.

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

Step 3.2

Multiply $- 1 (- \sqrt[3]{2})$ .

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Step 3.2.1

Multiply $- 1$ by $- 1$ .

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

Step 3.2.2

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

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

Step 3.3

Apply the product rule to $- \sqrt[3]{2}$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

$- (1 + \sqrt[3]{2} + \sqrt[3]{4})$

Step 4

Simplify by multiplying through.

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Step 4.1

Apply the distributive property.

$- 1 \cdot 1 - \sqrt[3]{2} - \sqrt[3]{4}$

Step 4.2

Multiply $- 1$ by $1$ .

$- 1 - \sqrt[3]{2} - \sqrt[3]{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- 1 - \sqrt[3]{2} - \sqrt[3]{4}$

Decimal Form:

$- 3.84732210 \dots$