Algebra Examples

$\frac{1}{20} + \frac{2}{5 x^{3}}$

Step 1

To write $\frac{1}{20}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\frac{1}{20} \cdot \frac{x^{3}}{x^{3}} + \frac{2}{5 x^{3}}$

Step 2

To write $\frac{2}{5 x^{3}}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{20} \cdot \frac{x^{3}}{x^{3}} + \frac{2}{5 x^{3}} \cdot \frac{4}{4}$

Step 3

Write each expression with a common denominator of $20 x^{3}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{20}$ by $\frac{x^{3}}{x^{3}}$ .

$\frac{x^{3}}{20 x^{3}} + \frac{2}{5 x^{3}} \cdot \frac{4}{4}$

Step 3.2

Multiply $\frac{2}{5 x^{3}}$ by $\frac{4}{4}$ .

$\frac{x^{3}}{20 x^{3}} + \frac{2 \cdot 4}{5 x^{3} \cdot 4}$

Step 3.3

Multiply $4$ by $5$ .

$\frac{x^{3}}{20 x^{3}} + \frac{2 \cdot 4}{20 x^{3}}$

Step 4

Combine the numerators over the common denominator.

$\frac{x^{3} + 2 \cdot 4}{20 x^{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $4$ .

$\frac{x^{3} + 8}{20 x^{3}}$

Step 5.2

Rewrite $x^{3} + 8$ in a factored form.

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Step 5.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{x^{3} + 2^{3}}{20 x^{3}}$

Step 5.2.2

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 $b = 2$ .

$\frac{(x + 2) (x^{2} - x \cdot 2 + 2^{2})}{20 x^{3}}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $2$ by $- 1$ .

$\frac{(x + 2) (x^{2} - 2 x + 2^{2})}{20 x^{3}}$

Step 5.2.3.2

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

$\frac{(x + 2) (x^{2} - 2 x + 4)}{20 x^{3}}$