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

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

Step 1

Multiply the numerator and denominator of the complex fraction by $b^{2}$ .

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $b$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $b^{2}$ .

Move the leading negative in $- \frac{a^{2}}{b^{2}}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

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

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

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

Factor $b$ out of $b a$ .

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

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

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

Multiply $b$ by $1$ .

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

Step 5

Simplify the denominator.

Rewrite $b^{2} \cdot 1$ as ${(b \cdot 1)}^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = b \cdot 1$ and $b = a$ .

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

Simplify.

Multiply $b$ by $1$ .

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

Multiply $b$ by $1$ .

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

Step 6

Cancel the common factor of $b + a$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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