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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $b$ .

Factor $b$ out of $b a$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

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

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

Factor $a$ out of $b a$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

Factor $a$ out of $b a$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

Simplify the denominator.

Factor $b$ out of $b a \cdot 1 + b^{2}$ .

Factor $b$ out of $b a \cdot 1$ .

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

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

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

Factor $b$ out of $b (a \cdot 1) + b \cdot b$ .

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

Multiply $a$ by $1$ .

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

Step 6

Cancel the common factor of $a + b$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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