Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify each term.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Reorder the factors of $b a$ .

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

Step 4.4

Combine the numerators over the common denominator.

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Combine fractions.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

Simplify the numerator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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Step 7.3.1

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

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

Step 7.3.2

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

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

Step 7.3.3

Reorder the factors of $b a$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 7.5

Combine exponents.

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Step 7.5.1

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

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

Step 7.5.2

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

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

Step 7.6

Reduce the expression $\frac{(b + a) a b}{a b}$ by cancelling the common factors.

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Step 7.6.1

Cancel the common factor.

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

Step 7.6.2

Rewrite the expression.

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

Step 7.7

Cancel the common factor of $b$ .

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Step 7.7.1

Cancel the common factor.

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

Step 7.7.2

Divide $b + a$ by $1$ .

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