Algebra Examples

$\frac{1}{8} x^{3} - 216$

Step 1

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

${(\frac{1}{2} x)}^{3} - 216$

Step 2

Rewrite $216$ as $6^{3}$ .

${(\frac{1}{2} x)}^{3} - 6^{3}$

Step 3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = \frac{1}{2} x$ and $b = 6$ .

$(\frac{1}{2} x - 6) ({(\frac{1}{2} x)}^{2} + \frac{1}{2} x \cdot 6 + 6^{2})$

Step 4

Simplify.

Tap for more steps...

Step 4.1

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

$(\frac{x}{2} - 6) ({(\frac{1}{2} x)}^{2} + \frac{1}{2} x \cdot 6 + 6^{2})$

Step 4.2

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

$(\frac{x}{2} - 6) ({(\frac{x}{2})}^{2} + \frac{1}{2} x \cdot 6 + 6^{2})$

Step 4.3

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

$(\frac{x}{2} - 6) (\frac{x^{2}}{2^{2}} + \frac{1}{2} x \cdot 6 + 6^{2})$

Step 4.4

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

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + \frac{1}{2} x \cdot 6 + 6^{2})$

Step 4.5

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

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + \frac{x}{2} \cdot 6 + 6^{2})$

Step 4.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.6.1

Factor $2$ out of $6$ .

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + \frac{x}{2} \cdot (2 (3)) + 6^{2})$

Step 4.6.2

Cancel the common factor.

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + \frac{x}{2} \cdot (2 \cdot 3) + 6^{2})$

Step 4.6.3

Rewrite the expression.

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + x \cdot 3 + 6^{2})$

Step 4.7

Move $3$ to the left of $x$ .

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + 3 x + 6^{2})$

Step 4.8

Raise $6$ to the power of $2$ .

$(\frac{x}{2} - 6) (\frac{x^{2}}{4} + 3 x + 36)$

Step 4.9

Factor.

Tap for more steps...

Step 4.9.1

Factor $\frac{1}{4}$ out of each term.

$(\frac{x}{2} - 6) (\frac{1}{4} (x^{2} + 12 x + 144))$

Step 4.9.2

Remove unnecessary parentheses.

$(\frac{x}{2} - 6) \frac{1}{4} (x^{2} + 12 x + 144)$