Basic Math Examples

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

Step 1

Simplify the numerator.

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Step 1.1

To write $\frac{1}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Step 1.2

To write $- \frac{1}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

Step 1.3

Write each expression with a common denominator of $a b$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.3.1

Multiply $\frac{1}{a}$ by $\frac{b}{b}$ .

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

Step 1.3.2

Multiply $\frac{1}{b}$ by $\frac{a}{a}$ .

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

Step 1.3.3

Reorder the factors of $b a$ .

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

To write $\frac{1}{b^{2}}$ as a fraction with a common denominator, multiply by $\frac{a^{2}}{a^{2}}$ .

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

Step 2.2

To write $- \frac{1}{a^{2}}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

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

Step 2.3

Write each expression with a common denominator of $b^{2} a^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{b^{2}}$ by $\frac{a^{2}}{a^{2}}$ .

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

Step 2.3.2

Multiply $\frac{1}{a^{2}}$ by $\frac{b^{2}}{b^{2}}$ .

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

Step 2.3.3

Reorder the factors of $b^{2} a^{2}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

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 $b = b$ .

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $a + b$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

Multiply $\frac{b - a}{a b}$ by $\frac{a^{2} b^{2}}{a - b}$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

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

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

Step 6.2.2

Cancel the common factor.

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

Step 6.2.3

Rewrite the expression.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 8

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

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Step 8.1

Factor $- 1$ out of $b$ .

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

Step 8.2

Factor $- 1$ out of $- a$ .

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

Step 8.3

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

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

Step 8.4

Reorder terms.

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

Step 8.5

Cancel the common factor.

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

Step 8.6

Divide $- 1 (a b)$ by $1$ .

$- 1 (a b)$

Step 9

Rewrite $- 1 (a b)$ as $- (a b)$ .

$- a b$