Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Reorder the factors of $b a$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

Step 6.2

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

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

Step 7

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

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

Step 8

Simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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Step 8.3.1

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

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

Step 8.3.2

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

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

Step 8.3.3

Reorder the factors of $b a$ .

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 9

Multiply the numerator by the reciprocal of the denominator.

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

Step 10

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

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

Step 11

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

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

Add $1$ and $1$ .

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

Step 11.6

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

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

Step 11.7

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

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

Step 11.8

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

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

Step 11.9

Add $1$ and $1$ .

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