Factor 1/(8x^6)+(y^3)/216

$\frac{1}{8 x^{6}} + \frac{y^{3}}{216}$

Step 1

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

$\frac{1^{3}}{8 x^{6}} + \frac{y^{3}}{216}$

Step 2

Rewrite $8 x^{6}$ as ${(2 x^{2})}^{3}$ .

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\frac{y^{3}}{6^{3}}$ as ${(\frac{y}{6})}^{3}$ .

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

Step 6

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 $a = \frac{1}{2 x^{2}}$ and $b = \frac{y}{6}$ .

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

Step 7

Simplify.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{1}{2 x^{2}}$ .

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

Apply the product rule to $2 x^{2}$ .

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

One to any power is one.

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

Simplify the denominator.

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

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply $- \frac{1}{2 x^{2}} \cdot \frac{y}{6}$ .

Multiply $\frac{y}{6}$ by $\frac{1}{2 x^{2}}$ .

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

Multiply $2$ by $6$ .

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

Apply the product rule to $\frac{y}{6}$ .

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

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

$(\frac{1}{2 x^{2}} + \frac{y}{6}) (\frac{1}{4 x^{4}} - \frac{y}{12 x^{2}} + \frac{y^{2}}{36})$

Factor.

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

$(\frac{1}{2 x^{2}} + \frac{y}{6}) (\frac{1}{36} (\frac{9}{x^{4}} - \frac{3 y}{x^{2}} + y^{2}))$

Remove unnecessary parentheses.

$(\frac{1}{2 x^{2}} + \frac{y}{6}) \frac{1}{36} (\frac{9}{x^{4}} - \frac{3 y}{x^{2}} + y^{2})$

Factor 1/(8x^6)+(y^3)/216

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