Simplify (1+ab^-1)/(1-ab^-1)

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

Simplify the numerator.

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Simplify the denominator.

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

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

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

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

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $b$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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