Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Cancel the common factor of $b$ .

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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