Solve (1/a-b-b/a^{2-b^2})diva^2-ab/a^2-2ab+b^2

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

Evaluate

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

Short Solution Steps

Factor a^{2}-b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and \left(a+b\right)\left(a-b\right) is \left(a+b\right)\left(a-b\right). Multiply \frac{1}{a-b} times \frac{a+b}{a+b}.

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

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

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

Combine like terms in a+b-b.

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

Factor the expressions that are not already factored in \frac{a^{2}-ab}{a^{2}-2ab+b^{2}}.

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

Cancel out a-b in both numerator and denominator.

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

Divide \frac{a}{\left(a+b\right)\left(a-b\right)} by \frac{a}{a-b} by multiplying \frac{a}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{a}{a-b}.

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

Cancel out a\left(a-b\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor a^{2}-b^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a-b and \left(a+b\right)\left(a-b\right) is \left(a+b\right)\left(a-b\right). Multiply \frac{1}{a-b} times \frac{a+b}{a+b}.

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

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

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

Combine like terms in a+b-b.

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

Factor the expressions that are not already factored in \frac{a^{2}-ab}{a^{2}-2ab+b^{2}}.

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

Cancel out a-b in both numerator and denominator.

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

Divide \frac{a}{\left(a+b\right)\left(a-b\right)} by \frac{a}{a-b} by multiplying \frac{a}{\left(a+b\right)\left(a-b\right)} by the reciprocal of \frac{a}{a-b}.

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

Cancel out a\left(a-b\right) in both numerator and denominator.

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