Algebra Examples

$\frac{1}{64} m^{3} + 1000$

Step 1

Rewrite $\frac{1}{64} m^{3}$ as ${(\frac{1}{4} m)}^{3}$ .

${(\frac{1}{4} m)}^{3} + 1000$

Step 2

Rewrite $1000$ as $10^{3}$ .

${(\frac{1}{4} m)}^{3} + 10^{3}$

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 $a = \frac{1}{4} m$ and $b = 10$ .

$(\frac{1}{4} m + 10) ({(\frac{1}{4} m)}^{2} - (\frac{1}{4} m) \cdot 10 + 10^{2})$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{4}$ and $m$ .

$(\frac{m}{4} + 10) ({(\frac{1}{4} m)}^{2} - (\frac{1}{4} m) \cdot 10 + 10^{2})$

Step 4.2

Combine $\frac{1}{4}$ and $m$ .

$(\frac{m}{4} + 10) ({(\frac{m}{4})}^{2} - (\frac{1}{4} m) \cdot 10 + 10^{2})$

Step 4.3

Apply the product rule to $\frac{m}{4}$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{4^{2}} - (\frac{1}{4} m) \cdot 10 + 10^{2})$

Step 4.4

Raise $4$ to the power of $2$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} - (\frac{1}{4} m) \cdot 10 + 10^{2})$

Step 4.5

Combine $\frac{1}{4}$ and $m$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} - \frac{m}{4} \cdot 10 + 10^{2})$

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Move the leading negative in $- \frac{m}{4}$ into the numerator.

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m}{4} \cdot 10 + 10^{2})$

Step 4.6.2

Factor $2$ out of $4$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m}{2 (2)} \cdot 10 + 10^{2})$

Step 4.6.3

Factor $2$ out of $10$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m}{2 \cdot 2} \cdot (2 \cdot 5) + 10^{2})$

Step 4.6.4

Cancel the common factor.

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m}{2 \cdot 2} \cdot (2 \cdot 5) + 10^{2})$

Step 4.6.5

Rewrite the expression.

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m}{2} \cdot 5 + 10^{2})$

Step 4.7

Combine $\frac{- m}{2}$ and $5$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- m \cdot 5}{2} + 10^{2})$

Step 4.8

Multiply $5$ by $- 1$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} + \frac{- 5 m}{2} + 10^{2})$

Step 4.9

Move the negative in front of the fraction.

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} - \frac{5 m}{2} + 10^{2})$

Step 4.10

Raise $10$ to the power of $2$ .

$(\frac{m}{4} + 10) (\frac{m^{2}}{16} - \frac{5 m}{2} + 100)$

Step 4.11

Factor.

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Step 4.11.1

Factor $\frac{1}{16}$ out of each term.

$(\frac{m}{4} + 10) (\frac{1}{16} (m^{2} - 40 m + 1600))$

Step 4.11.2

Remove unnecessary parentheses.

$(\frac{m}{4} + 10) \frac{1}{16} (m^{2} - 40 m + 1600)$