Pre-Algebra Examples

$\frac{5 a^{- 1} + 4 b^{- 2}}{a^{- 1} - b^{- 1}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{5 \frac{1}{a} + 4 b^{- 2}}{a^{- 1} - b^{- 1}}$

Step 1.2

Combine $5$ and $\frac{1}{a}$ .

$\frac{\frac{5}{a} + 4 b^{- 2}}{a^{- 1} - b^{- 1}}$

Step 1.3

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

$\frac{\frac{5}{a} + 4 \frac{1}{b^{2}}}{a^{- 1} - b^{- 1}}$

Step 1.4

Combine $4$ and $\frac{1}{b^{2}}$ .

$\frac{\frac{5}{a} + \frac{4}{b^{2}}}{a^{- 1} - b^{- 1}}$

Step 1.5

To write $\frac{5}{a}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

$\frac{\frac{5}{a} \cdot \frac{b^{2}}{b^{2}} + \frac{4}{b^{2}}}{a^{- 1} - b^{- 1}}$

Step 1.6

To write $\frac{4}{b^{2}}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$\frac{\frac{5}{a} \cdot \frac{b^{2}}{b^{2}} + \frac{4}{b^{2}} \cdot \frac{a}{a}}{a^{- 1} - b^{- 1}}$

Step 1.7

Write each expression with a common denominator of $a b^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.7.1

Multiply $\frac{5}{a}$ by $\frac{b^{2}}{b^{2}}$ .

$\frac{\frac{5 b^{2}}{a b^{2}} + \frac{4}{b^{2}} \cdot \frac{a}{a}}{a^{- 1} - b^{- 1}}$

Step 1.7.2

Multiply $\frac{4}{b^{2}}$ by $\frac{a}{a}$ .

$\frac{\frac{5 b^{2}}{a b^{2}} + \frac{4 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

Step 1.7.3

Reorder the factors of $a b^{2}$ .

$\frac{\frac{5 b^{2}}{b^{2} a} + \frac{4 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

Step 1.8

Combine the numerators over the common denominator.

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{1}{a} - b^{- 1}}$

Step 2.2

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{1}{a} - \frac{1}{b}}$

Step 2.3

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{1}{a} \cdot \frac{b}{b} - \frac{1}{b}}$

Step 2.4

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{1}{a} \cdot \frac{b}{b} - \frac{1}{b} \cdot \frac{a}{a}}$

Step 2.5

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

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Step 2.5.1

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{b}{a b} - \frac{1}{b} \cdot \frac{a}{a}}$

Step 2.5.2

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

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{b}{a b} - \frac{a}{b a}}$

Step 2.5.3

Reorder the factors of $b a$ .

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{b}{a b} - \frac{a}{a b}}$

Step 2.6

Combine the numerators over the common denominator.

$\frac{\frac{5 b^{2} + 4 a}{b^{2} a}}{\frac{b - a}{a b}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 b^{2} + 4 a}{b^{2} a} \cdot \frac{a b}{b - a}$

Step 4

Cancel the common factor of $a b$ .

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Step 4.1

Factor $a b$ out of $b^{2} a$ .

$\frac{5 b^{2} + 4 a}{a b (b)} \cdot \frac{a b}{b - a}$

Step 4.2

Cancel the common factor.

$\frac{5 b^{2} + 4 a}{a b \cdot b} \cdot \frac{a b}{b - a}$

Step 4.3

Rewrite the expression.

$\frac{5 b^{2} + 4 a}{b} \cdot \frac{1}{b - a}$

Step 5

Multiply $\frac{5 b^{2} + 4 a}{b}$ by $\frac{1}{b - a}$ .

$\frac{5 b^{2} + 4 a}{b (b - a)}$