Algebra Examples

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

Step 1

To write $x^{3}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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 $a = x \cdot x$ and $b = 1$ .

$\frac{(x \cdot x + 1) ({(x \cdot x)}^{2} - (x \cdot x) \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $x$ by $x$ .

$\frac{(x^{2} + 1) ({(x \cdot x)}^{2} - (x \cdot x) \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4.2

Multiply $x$ by $x$ .

$\frac{(x^{2} + 1) ({(x^{2})}^{2} - (x \cdot x) \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4.3

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.4.3.1

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

$\frac{(x^{2} + 1) (x^{2 \cdot 2} - (x \cdot x) \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4.3.2

Multiply $2$ by $2$ .

$\frac{(x^{2} + 1) (x^{4} - (x \cdot x) \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4.4

Multiply $x$ by $x$ .

$\frac{(x^{2} + 1) (x^{4} - x^{2} \cdot 1 + 1^{2})}{x^{3}}$

Step 3.4.5

Simplify $- x^{2} \cdot 1$ .

$\frac{(x^{2} + 1) (x^{4} - x^{2} + 1^{2})}{x^{3}}$

Step 3.4.6

One to any power is one.

$\frac{(x^{2} + 1) (x^{4} - x^{2} + 1)}{x^{3}}$