Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

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

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

Step 4

Combine.

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

Step 5

Multiply $1$ by $1$ .

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

Step 6

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

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

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 $\frac{b}{b}$ .

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

Step 7.2

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

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

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 $\frac{b}{b}$ .

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

Step 7.3.2

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

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

Step 7.3.3

Reorder the factors of $b a$ .

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

Step 7.4

Combine the numerators over the common denominator.

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

Step 8

Multiply the numerator by the reciprocal of the denominator.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

Add $2$ and $1$ .

$\frac{b + a}{a^{3} b \cdot b^{2}}$

Step 9.5

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

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Step 9.5.1

Move $b^{2}$ .

$\frac{b + a}{a^{3} (b^{2} b)}$

Step 9.5.2

Multiply $b^{2}$ by $b$ .

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Step 9.5.2.1

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

$\frac{b + a}{a^{3} (b^{2} b^{1})}$

Step 9.5.2.2

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

$\frac{b + a}{a^{3} b^{2 + 1}}$

Step 9.5.3

Add $2$ and $1$ .

$\frac{b + a}{a^{3} b^{3}}$