Algebra Examples

$\frac{1}{8} x^{3} + \frac{8}{27}$

Step 1

Rewrite $\frac{1}{8} x^{3}$ as ${(\frac{1}{2} x)}^{3}$ .

${(\frac{1}{2} x)}^{3} + \frac{8}{27}$

Step 2

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

${(\frac{1}{2} x)}^{3} + \frac{2^{3}}{27}$

Step 3

Rewrite $27$ as $3^{3}$ .

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

Step 4

Rewrite $\frac{2^{3}}{3^{3}}$ as ${(\frac{2}{3})}^{3}$ .

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

Step 5

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 $a = \frac{1}{2} x$ and $b = \frac{2}{3}$ .

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

Step 6

Simplify.

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Step 6.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 6.2

Combine $\frac{1}{2}$ and $x$ .

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

Step 6.3

Apply the product rule to $\frac{x}{2}$ .

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

Step 6.4

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

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

Step 6.5

Combine $\frac{1}{2}$ and $x$ .

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

Step 6.6

Cancel the common factor of $2$ .

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Step 6.6.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

Step 6.6.2

Cancel the common factor.

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

Step 6.6.3

Rewrite the expression.

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

Step 6.7

Combine $\frac{1}{3}$ and $x$ .

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

Step 6.8

Apply the product rule to $\frac{2}{3}$ .

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

Step 6.9

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

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

Step 6.10

Raise $3$ to the power of $2$ .

$(\frac{x}{2} + \frac{2}{3}) (\frac{x^{2}}{4} - \frac{x}{3} + \frac{4}{9})$

Step 6.11

Factor.

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Step 6.11.1

Factor $\frac{1}{36}$ out of each term.

$(\frac{x}{2} + \frac{2}{3}) (\frac{1}{36} (9 x^{2} - 12 x + 16))$

Step 6.11.2

Remove unnecessary parentheses.

$(\frac{x}{2} + \frac{2}{3}) \frac{1}{36} (9 x^{2} - 12 x + 16)$