Simplify (b/a-a/b)/(1/a-1/b)

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

Step 1

Multiply the numerator and denominator of the complex fraction by $a b$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $a$ .

Factor $a$ out of $a b$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $b$ .

Move the leading negative in $- \frac{a}{b}$ into the numerator.

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

Factor $b$ out of $a b$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

Factor $a$ out of $a b$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $b$ .

Move the leading negative in $- \frac{1}{b}$ into the numerator.

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

Factor $b$ out of $a b$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

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

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

Step 5

Simplify the denominator.

Move $- 1$ to the left of $a$ .

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

Rewrite $- 1 a$ as $- a$ .

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

Step 6

Cancel the common factor of $b - a$ .

Cancel the common factor.

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

Divide $b + a$ by $1$ .

$b + a$

Simplify (b/a-a/b)/(1/a-1/b)

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