Algebra Examples

$x^{3} + \frac{1}{x^{3}}$

Step 1

Rewrite $1$ as $1^{3}$ .

$x^{3} + \frac{1^{3}}{x^{3}}$

Step 2

Rewrite $\frac{1^{3}}{x^{3}}$ as ${(\frac{1}{x})}^{3}$ .

$x^{3} + {(\frac{1}{x})}^{3}$

Step 3

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = \frac{1}{x}$ .

$(x + \frac{1}{x}) (x^{2} - x \frac{1}{x} + {(\frac{1}{x})}^{2})$

Step 4

Simplify.

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Step 4.1

Cancel the common factor of $x$ .

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Step 4.1.1

Factor $x$ out of $- x$ .

$(x + \frac{1}{x}) (x^{2} + x \cdot - 1 \frac{1}{x} + {(\frac{1}{x})}^{2})$

Step 4.1.2

Cancel the common factor.

$(x + \frac{1}{x}) (x^{2} + x \cdot - 1 \frac{1}{x} + {(\frac{1}{x})}^{2})$

Step 4.1.3

Rewrite the expression.

$(x + \frac{1}{x}) (x^{2} - 1 + {(\frac{1}{x})}^{2})$

Step 4.2

Apply the product rule to $\frac{1}{x}$ .

$(x + \frac{1}{x}) (x^{2} - 1 + \frac{1^{2}}{x^{2}})$

Step 4.3

One to any power is one.

$(x + \frac{1}{x}) (x^{2} - 1 + \frac{1}{x^{2}})$

Step 4.4

Reorder terms.

$(x + \frac{1}{x}) (x^{2} + \frac{1}{x^{2}} - 1)$