Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify.

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

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

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

Step 1.4.5

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

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Step 1.4.5.1

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

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

Step 1.4.5.2

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

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

Step 1.4.5.3

Reorder the factors of $b a$ .

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

Step 1.4.6

Combine the numerators over the common denominator.

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

Step 1.4.7

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

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

Step 1.4.8

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

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

Step 1.4.9

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

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

Step 1.4.10

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

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

Step 1.4.11

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

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Step 1.4.11.1

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

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

Step 1.4.11.2

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

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

Step 1.4.11.3

Reorder the factors of $b a$ .

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

Step 1.4.12

Combine the numerators over the common denominator.

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

Step 2

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

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $1$ .

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

Add $1$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine.

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

Step 6

Multiply $a^{2}$ by $a^{2}$ by adding the exponents.

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Step 6.1

Move $a^{2}$ .

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

Step 6.2

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

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

Step 6.3

Add $2$ and $2$ .

$\frac{(b + a) (b - a) \cdot 1}{a^{4} b^{2} b^{2}}$

Step 7

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

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Step 7.1

Move $b^{2}$ .

$\frac{(b + a) (b - a) \cdot 1}{a^{4} (b^{2} b^{2})}$

Step 7.2

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

$\frac{(b + a) (b - a) \cdot 1}{a^{4} b^{2 + 2}}$

Step 7.3

Add $2$ and $2$ .

$\frac{(b + a) (b - a) \cdot 1}{a^{4} b^{4}}$

Step 8

Multiply $(b + a) (b - a)$ by $1$ .

$\frac{(b + a) (b - a)}{a^{4} b^{4}}$