Basic Math Examples

$\frac{a^{5} - a^{2}}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $a^{2}$ out of $a^{5} - a^{2}$ .

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Step 1.1.1

Factor $a^{2}$ out of $a^{5}$ .

$\frac{a^{2} a^{3} - a^{2}}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.1.2

Factor $a^{2}$ out of $- a^{2}$ .

$\frac{a^{2} a^{3} + a^{2} \cdot - 1}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.1.3

Factor $a^{2}$ out of $a^{2} a^{3} + a^{2} \cdot - 1$ .

$\frac{a^{2} (a^{3} - 1)}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.2

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

$\frac{a^{2} (a^{3} - 1^{3})}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.3

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 $a = a$ and $b = 1$ .

$\frac{a^{2} ((a - 1) (a^{2} + a \cdot 1 + 1^{2}))}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $a$ by $1$ .

$\frac{a^{2} ((a - 1) (a^{2} + a + 1^{2}))}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 1.4.2

One to any power is one.

$\frac{a^{2} ((a - 1) (a^{2} + a + 1))}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

$\frac{a^{2} (a - 1) (a^{2} + a + 1)}{b^{3}} \cdot \frac{c^{4}}{a^{2}}$

Step 2

Simplify terms.

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Step 2.1

Combine.

$\frac{a^{2} (a - 1) (a^{2} + a + 1) c^{4}}{b^{3} a^{2}}$

Step 2.2

Cancel the common factor of $a^{2}$ .

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Step 2.2.1

Cancel the common factor.

$\frac{a^{2} (a - 1) (a^{2} + a + 1) c^{4}}{b^{3} a^{2}}$

Step 2.2.2

Rewrite the expression.

$\frac{((a - 1) (a^{2} + a + 1)) c^{4}}{b^{3}}$

Step 2.3

Reorder factors in $\frac{(a - 1) (a^{2} + a + 1) c^{4}}{b^{3}}$ .

$\frac{c^{4} (a - 1) (a^{2} + a + 1)}{b^{3}}$