Solve frac{a^{-1}}{a^{-1}+b^-1}+a^-1/a^-1-b^-1=

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

Evaluate

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

Short Solution Steps

Factor the expressions that are not already factored in \frac{a^{-1}}{a^{-1}+b^{-1}}.

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

Cancel out \frac{1}{a} in both numerator and denominator.

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

Express \frac{1}{b}a as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b}{b}.

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

Since \frac{b}{b} and \frac{a}{b} have the same denominator, add them by adding their numerators.

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

Divide 1 by \frac{b+a}{b} by multiplying 1 by the reciprocal of \frac{b+a}{b}.

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

Factor the expressions that are not already factored in \frac{a^{-1}}{a^{-1}-b^{-1}}.

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

Cancel out \frac{1}{a} in both numerator and denominator.

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

Express \frac{1}{b}a as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{b}{b}.

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

Since -\frac{a}{b} and \frac{b}{b} have the same denominator, add them by adding their numerators.

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

Divide 1 by \frac{-a+b}{b} by multiplying 1 by the reciprocal of \frac{-a+b}{b}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of b+a and -a+b is \left(a+b\right)\left(-a+b\right). Multiply \frac{b}{b+a} times \frac{-a+b}{-a+b}. Multiply \frac{b}{-a+b} times \frac{a+b}{a+b}.

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

Since \frac{b\left(-a+b\right)}{\left(a+b\right)\left(-a+b\right)} and \frac{b\left(a+b\right)}{\left(a+b\right)\left(-a+b\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in b\left(-a+b\right)+b\left(a+b\right).

$$\frac{-ba+b^{2}+ba+b^{2}}{\left(a+b\right)\left(-a+b\right)}$$

Combine like terms in -ba+b^{2}+ba+b^{2}.

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

Expand \left(a+b\right)\left(-a+b\right).

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

Factor

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