Simplify (3a^-1+5b^-2)/(a^-1-b^-1)

$\frac{3 a^{- 1} + 5 b^{- 2}}{a^{- 1} - b^{- 1}}$

Step 1

Simplify the numerator.

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

$\frac{3 \frac{1}{a} + 5 b^{- 2}}{a^{- 1} - b^{- 1}}$

Combine $3$ and $\frac{1}{a}$ .

$\frac{\frac{3}{a} + 5 b^{- 2}}{a^{- 1} - b^{- 1}}$

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

$\frac{\frac{3}{a} + 5 \frac{1}{b^{2}}}{a^{- 1} - b^{- 1}}$

Combine $5$ and $\frac{1}{b^{2}}$ .

$\frac{\frac{3}{a} + \frac{5}{b^{2}}}{a^{- 1} - b^{- 1}}$

To write $\frac{3}{a}$ as a fraction with a common denominator, multiply by $\frac{b^{2}}{b^{2}}$ .

$\frac{\frac{3}{a} \cdot \frac{b^{2}}{b^{2}} + \frac{5}{b^{2}}}{a^{- 1} - b^{- 1}}$

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

$\frac{\frac{3}{a} \cdot \frac{b^{2}}{b^{2}} + \frac{5}{b^{2}} \cdot \frac{a}{a}}{a^{- 1} - b^{- 1}}$

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

Multiply $\frac{3}{a}$ by $\frac{b^{2}}{b^{2}}$ .

$\frac{\frac{3 b^{2}}{a b^{2}} + \frac{5}{b^{2}} \cdot \frac{a}{a}}{a^{- 1} - b^{- 1}}$

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

$\frac{\frac{3 b^{2}}{a b^{2}} + \frac{5 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

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

$\frac{\frac{3 b^{2}}{b^{2} a} + \frac{5 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

Combine the numerators over the common denominator.

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{a^{- 1} - b^{- 1}}$

Step 2

Simplify the denominator.

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{1}{a} - b^{- 1}}$

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{1}{a} - \frac{1}{b}}$

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{1}{a} \cdot \frac{b}{b} - \frac{1}{b}}$

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{1}{a} \cdot \frac{b}{b} - \frac{1}{b} \cdot \frac{a}{a}}$

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

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{b}{a b} - \frac{1}{b} \cdot \frac{a}{a}}$

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

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{b}{a b} - \frac{a}{b a}}$

Reorder the factors of $b a$ .

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{b}{a b} - \frac{a}{a b}}$

Combine the numerators over the common denominator.

$\frac{\frac{3 b^{2} + 5 a}{b^{2} a}}{\frac{b - a}{a b}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 b^{2} + 5 a}{b^{2} a} \cdot \frac{a b}{b - a}$

Step 4

Cancel the common factor of $a b$ .

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

$\frac{3 b^{2} + 5 a}{a b (b)} \cdot \frac{a b}{b - a}$

Cancel the common factor.

$\frac{3 b^{2} + 5 a}{a b \cdot b} \cdot \frac{a b}{b - a}$

Rewrite the expression.

$\frac{3 b^{2} + 5 a}{b} \cdot \frac{1}{b - a}$

Step 5

Multiply $\frac{3 b^{2} + 5 a}{b}$ by $\frac{1}{b - a}$ .

$\frac{3 b^{2} + 5 a}{b (b - a)}$

Simplify (3a^-1+5b^-2)/(a^-1-b^-1)

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