Precalculus Examples

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

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 $b = b$ .

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

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 2 b \frac{a - b}{a + b}$ .

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Step 2.1.1

Combine $- 2$ and $\frac{a - b}{a + b}$ .

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

Step 2.1.2

Combine $b$ and $\frac{- 2 (a - b)}{a + b}$ .

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

Step 2.2

Remove unnecessary parentheses.

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

Step 2.3

Move $- 2$ to the left of $b$ .

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

Step 2.4

Move the negative in front of the fraction.

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

Step 2.5

To write $a$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

To write $- b$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

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

Step 2.8

Combine $- b$ and $\frac{a + b}{a + b}$ .

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 2.10

Rewrite $\frac{- b (a + b) + a (a + b) - 2 b (a - b)}{a + b}$ in a factored form.

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Step 2.10.1

Apply the distributive property.

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

Step 2.10.2

Multiply $b$ by $b$ by adding the exponents.

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Step 2.10.2.1

Move $b$ .

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

Step 2.10.2.2

Multiply $b$ by $b$ .

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

Step 2.10.3

Apply the distributive property.

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

Step 2.10.4

Multiply $a$ by $a$ .

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

Step 2.10.5

Apply the distributive property.

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

Step 2.10.6

Rewrite using the commutative property of multiplication.

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

Step 2.10.7

Simplify each term.

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Step 2.10.7.1

Multiply $b$ by $b$ by adding the exponents.

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Step 2.10.7.1.1

Move $b$ .

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

Step 2.10.7.1.2

Multiply $b$ by $b$ .

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

Step 2.10.7.2

Multiply $- 2$ by $- 1$ .

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

Step 2.10.8

Add $- b a$ and $a b$ .

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Step 2.10.8.1

Move $b$ .

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

Step 2.10.8.2

Add $- a b$ and $a b$ .

$\frac{a}{(a + b) (a - b)} \cdot \frac{\frac{- b^{2} + a^{2} + 0 - 2 b a + 2 b^{2}}{a + b}}{\frac{a - b}{b} \cdot \frac{a}{a + b}}$

Step 2.10.9

Add $- b^{2}$ and $0$ .

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

Step 2.10.10

Add $- b^{2}$ and $2 b^{2}$ .

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

Step 2.10.11

Factor using the perfect square rule.

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Step 2.10.11.1

Rearrange terms.

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

Step 2.10.11.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 b a = 2 \cdot a \cdot b$

Step 2.10.11.3

Rewrite the polynomial.

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

Step 2.10.11.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = a$ and $b = b$ .

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

Step 3

Multiply $\frac{a - b}{b}$ by $\frac{a}{a + b}$ .

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

Step 4

Combine fractions.

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Step 4.1

Combine.

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

Step 4.2

Combine $a$ and $\frac{{(a - b)}^{2}}{a + b}$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $a - b$ .

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Step 6.1

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

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

Step 6.2

Factor $a - b$ out of $(a + b) (a - b) \frac{(a - b) a}{b (a + b)}$ .

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

Step 6.3

Cancel the common factor.

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

Step 6.4

Rewrite the expression.

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

Step 7

Multiply $\frac{a (a - b)}{a + b}$ by $\frac{1}{(a + b) \frac{(a - b) a}{b (a + b)}}$ .

$\frac{a (a - b)}{(a + b) ((a + b) \frac{(a - b) a}{b (a + b)})}$

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Add $1$ and $1$ .

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

Step 12

Combine ${(a + b)}^{2}$ and $\frac{(a - b) a}{b (a + b)}$ .

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

Step 13

Cancel the common factor of ${(a + b)}^{2}$ and $a + b$ .

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Step 13.1

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

$\frac{a (a - b)}{\frac{(a + b) ((a + b) ((a - b) a))}{b (a + b)}}$

Step 13.2

Cancel the common factors.

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Step 13.2.1

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

$\frac{a (a - b)}{\frac{(a + b) ((a + b) ((a - b) a))}{(a + b) b}}$

Step 13.2.2

Cancel the common factor.

$\frac{a (a - b)}{\frac{(a + b) ((a + b) ((a - b) a))}{(a + b) b}}$

Step 13.2.3

Rewrite the expression.

$\frac{a (a - b)}{\frac{(a + b) ((a - b) a)}{b}}$

Step 14

Multiply the numerator by the reciprocal of the denominator.

$a (a - b) \frac{b}{(a + b) (a - b) a}$

Step 15

Cancel the common factor of $(a - b) a$ .

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

Cancel the common factor.

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

Step 15.4

Rewrite the expression.

$\frac{b}{a + b}$