Algebra Examples

$\frac{8}{27} + m^{6}$

Step 1

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

$\frac{2^{3}}{27} + m^{6}$

Step 2

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

$\frac{2^{3}}{3^{3}} + m^{6}$

Step 3

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

${(\frac{2}{3})}^{3} + m^{6}$

Step 4

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

${(\frac{2}{3})}^{3} + {(m^{2})}^{3}$

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 $a = \frac{2}{3}$ and $b = m^{2}$ .

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

Combine $m^{2}$ and $\frac{2}{3}$ .

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

Step 6.5

Move $2$ to the left of $m^{2}$ .

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

Step 6.6

Multiply $2$ by $m^{2}$ .

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

Step 6.7

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

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Step 6.7.1

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

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

Step 6.7.2

Multiply $2$ by $2$ .

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