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

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

Evaluate

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

Short Solution Steps

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

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

Cancel out a-b in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2ab}{2ab}.

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

Since \frac{2ab}{2ab} and \frac{a^{2}+b^{2}}{2ab} have the same denominator, add them by adding their numerators.

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

Divide \frac{a+b}{ab} by \frac{2ab+a^{2}+b^{2}}{2ab} by multiplying \frac{a+b}{ab} by the reciprocal of \frac{2ab+a^{2}+b^{2}}{2ab}.

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a+b in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

Cancel out a-b in both numerator and denominator.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2ab}{2ab}.

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

Since \frac{2ab}{2ab} and \frac{a^{2}+b^{2}}{2ab} have the same denominator, add them by adding their numerators.

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

Divide \frac{a+b}{ab} by \frac{2ab+a^{2}+b^{2}}{2ab} by multiplying \frac{a+b}{ab} by the reciprocal of \frac{2ab+a^{2}+b^{2}}{2ab}.

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out a+b in both numerator and denominator.

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