Pre-Algebra Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = b$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Factor $a b$ out of $a b (a) + a b (- 1 b)$ .

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

Step 2.2

Rewrite $- 1 b$ as $- b$ .

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

Step 3

Combine.

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

Step 4

Multiply $a$ by $a$ by adding the exponents.

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Step 4.1

Move $a$ .

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

Step 4.2

Multiply $a$ by $a$ .

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

Step 5

Multiply $b^{3}$ by $b$ by adding the exponents.

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Step 5.1

Move $b$ .

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

Step 5.2

Multiply $b$ by $b^{3}$ .

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Step 5.2.1

Raise $b$ to the power of $1$ .

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

Step 5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3

Add $1$ and $3$ .

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

Step 6

Cancel the common factor of $a - b$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{(a + b) (a^{4} b^{2})}{a^{2} b^{4}}$

Step 7

Cancel the common factor of $a^{4}$ and $a^{2}$ .

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Step 7.1

Factor $a^{2}$ out of $(a + b) (a^{4} b^{2})$ .

$\frac{a^{2} ((a + b) (a^{2} b^{2}))}{a^{2} b^{4}}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $a^{2}$ out of $a^{2} b^{4}$ .

$\frac{a^{2} ((a + b) (a^{2} b^{2}))}{a^{2} (b^{4})}$

Step 7.2.2

Cancel the common factor.

$\frac{a^{2} ((a + b) (a^{2} b^{2}))}{a^{2} b^{4}}$

Step 7.2.3

Rewrite the expression.

$\frac{(a + b) (a^{2} b^{2})}{b^{4}}$

Step 8

Cancel the common factor of $b^{2}$ and $b^{4}$ .

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Step 8.1

Factor $b^{2}$ out of $(a + b) (a^{2} b^{2})$ .

$\frac{b^{2} ((a + b) (a^{2}))}{b^{4}}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $b^{2}$ out of $b^{4}$ .

$\frac{b^{2} ((a + b) (a^{2}))}{b^{2} b^{2}}$

Step 8.2.2

Cancel the common factor.

$\frac{b^{2} ((a + b) (a^{2}))}{b^{2} b^{2}}$

Step 8.2.3

Rewrite the expression.

$\frac{(a + b) (a^{2})}{b^{2}}$

Step 9

Reorder factors in $\frac{(a + b) (a^{2})}{b^{2}}$ .

$\frac{a^{2} (a + b)}{b^{2}}$