Algebra Examples

$\frac{1}{125} x^{3} + 27 y^{3}$

Step 1

Rewrite $\frac{1}{125} x^{3}$ as ${(\frac{1}{5} x)}^{3}$ .

${(\frac{1}{5} x)}^{3} + 27 y^{3}$

Step 2

Rewrite $27 y^{3}$ as ${(3 y)}^{3}$ .

${(\frac{1}{5} x)}^{3} + {(3 y)}^{3}$

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 $a = \frac{1}{5} x$ and $b = 3 y$ .

$(\frac{1}{5} x + 3 y) ({(\frac{1}{5} x)}^{2} - (\frac{1}{5} x) (3 y) + {(3 y)}^{2})$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{5}$ and $x$ .

$(\frac{x}{5} + 3 y) ({(\frac{1}{5} x)}^{2} - (\frac{1}{5} x) (3 y) + {(3 y)}^{2})$

Step 4.2

Combine $\frac{1}{5}$ and $x$ .

$(\frac{x}{5} + 3 y) ({(\frac{x}{5})}^{2} - (\frac{1}{5} x) (3 y) + {(3 y)}^{2})$

Step 4.3

Apply the product rule to $\frac{x}{5}$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{5^{2}} - (\frac{1}{5} x) (3 y) + {(3 y)}^{2})$

Step 4.4

Raise $5$ to the power of $2$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - (\frac{1}{5} x) (3 y) + {(3 y)}^{2})$

Step 4.5

Combine $\frac{1}{5}$ and $x$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - \frac{x}{5} (3 y) + {(3 y)}^{2})$

Step 4.6

Multiply $- \frac{x}{5} (3 y)$ .

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Step 4.6.1

Multiply $3$ by $- 1$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - 3 \frac{x}{5} y + {(3 y)}^{2})$

Step 4.6.2

Combine $- 3$ and $\frac{x}{5}$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} + \frac{- 3 x}{5} y + {(3 y)}^{2})$

Step 4.6.3

Combine $\frac{- 3 x}{5}$ and $y$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} + \frac{- 3 x y}{5} + {(3 y)}^{2})$

Step 4.7

Move the negative in front of the fraction.

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - \frac{3 x y}{5} + {(3 y)}^{2})$

Step 4.8

Apply the product rule to $3 y$ .

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - \frac{3 x y}{5} + 3^{2} y^{2})$

Step 4.9

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

$(\frac{x}{5} + 3 y) (\frac{x^{2}}{25} - \frac{3 x y}{5} + 9 y^{2})$

Step 4.10

Factor.

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Step 4.10.1

Factor $\frac{1}{25}$ out of each term.

$(\frac{x}{5} + 3 y) (\frac{1}{25} (x^{2} - 15 x y + 225 y^{2}))$

Step 4.10.2

Remove unnecessary parentheses.

$(\frac{x}{5} + 3 y) \frac{1}{25} (x^{2} - 15 x y + 225 y^{2})$