Basic Math Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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 1.1.2

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 1.2

Simplify each term.

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Step 1.2.1

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 1.2.2

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 2

Expand $(\frac{1}{a} + \frac{1}{b}) (\frac{1}{a} - \frac{1}{b})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify terms.

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Step 3.1

Combine the opposite terms in $\frac{1}{a} \cdot \frac{1}{a} + \frac{1}{a} (- \frac{1}{b}) + \frac{1}{b} \cdot \frac{1}{a} + \frac{1}{b} (- \frac{1}{b})$ .

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Step 3.1.1

Reorder the factors in the terms $\frac{1}{a} (- \frac{1}{b})$ and $\frac{1}{b} \cdot \frac{1}{a}$ .

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

Step 3.1.2

Add $- \frac{1}{a} \cdot \frac{1}{b}$ and $\frac{1}{a} \cdot \frac{1}{b}$ .

${(\frac{1}{a} \cdot \frac{1}{a} + 0 + \frac{1}{b} (- \frac{1}{b}))}^{- 1}$

Step 3.1.3

Add $\frac{1}{a} \cdot \frac{1}{a}$ and $0$ .

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

Step 3.2

Simplify each term.

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Step 3.2.1

Multiply $\frac{1}{a} \cdot \frac{1}{a}$ .

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Step 3.2.1.1

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

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

Step 3.2.1.2

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

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

Step 3.2.1.3

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

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

Step 3.2.1.4

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

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

Step 3.2.1.5

Add $1$ and $1$ .

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

Step 3.2.2

Rewrite using the commutative property of multiplication.

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

Step 3.2.3

Multiply $- \frac{1}{b} \cdot \frac{1}{b}$ .

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

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

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

Step 3.2.3.5

Add $1$ and $1$ .

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

Step 4

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

$\frac{1}{\frac{1}{a^{2}} - \frac{1}{b^{2}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\frac{1}{\frac{1^{2}}{a^{2}} - \frac{1}{b^{2}}}$

Step 5.2

Rewrite $\frac{1^{2}}{a^{2}}$ as ${(\frac{1}{a})}^{2}$ .

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

Step 5.3

Rewrite $\frac{1}{b^{2}}$ as ${(\frac{1}{b})}^{2}$ .

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

Step 5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{a}$ and $b = \frac{1}{b}$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

Reorder the factors of $b a$ .

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

Step 5.8

Combine the numerators over the common denominator.

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

Step 5.9

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

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

Step 5.10

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

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

Step 5.11

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

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Step 5.11.1

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

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

Step 5.11.2

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

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

Step 5.11.3

Reorder the factors of $b a$ .

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

Step 5.12

Combine the numerators over the common denominator.

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

Step 6

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

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

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 7.4

Add $1$ and $1$ .

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Add $1$ and $1$ .

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

Step 8

Multiply the numerator by the reciprocal of the denominator.

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

Step 9

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

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