Solve (a-b/a+b+a+b/a-b)*(a^2+b^2/2ab+1)*ab/a^2+b^2=

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

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Multiply \frac{2a^{2}+2b^{2}}{\left(a+b\right)\left(a-b\right)} times \frac{a^{2}+b^{2}+2ab}{2ab} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(2a^{2}+2b^{2}\right)\left(a^{2}+b^{2}+2ab\right)}{\left(a+b\right)\left(a-b\right)\times 2ab} times \frac{ab}{a^{2}+b^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out 2\left(a+b\right)\left(a^{2}+b^{2}\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

Multiply \frac{2a^{2}+2b^{2}}{\left(a+b\right)\left(a-b\right)} times \frac{a^{2}+b^{2}+2ab}{2ab} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \frac{\left(2a^{2}+2b^{2}\right)\left(a^{2}+b^{2}+2ab\right)}{\left(a+b\right)\left(a-b\right)\times 2ab} times \frac{ab}{a^{2}+b^{2}} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out ab in both numerator and denominator.

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

Factor the expressions that are not already factored.

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

Cancel out 2\left(a+b\right)\left(a^{2}+b^{2}\right) in both numerator and denominator.

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