Algebra Examples

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

Step 1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

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 $b = b$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Factor $b$ out of $- a b$ .

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

Step 3.1.3

Factor $b$ out of $b \cdot b + b (- a)$ .

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

Step 3.2

Combine.

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

Step 4

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

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Step 4.1

Move $b$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $1$ and $3$ .

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

Step 5

Cancel the common factor of $a - b$ and $b - a$ .

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Step 5.1

Factor $- 1$ out of $a$ .

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

Step 5.2

Factor $- 1$ out of $- b$ .

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

Step 5.3

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

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

Step 5.4

Reorder terms.

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

Step 5.5

Cancel the common factor.

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

Step 5.6

Rewrite the expression.

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

Step 6

Factor out negative.

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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