Solve (x-2y/x+2y+frac{x+2y/x-2y)*(1+frac{x^{2}+4y^2}{4xy})}{x^2+4y^2/2xy*(x^2+2xy)}

$$\frac { ( \frac { x - 2 y } { x + 2 y } + \frac { x + 2 y } { x - 2 y } ) \cdot ( 1 + \frac { x ^ { 2 } + 4 y ^ { 2 } } { 4 x y } ) } { \frac { x ^ { 2 } + 4 y ^ { 2 } } { 2 x y } \cdot ( x ^ { 2 } + 2 x y ) }$$

Evaluate

$$\frac{1}{x\left(x-2y\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2y and x-2y is \left(x-2y\right)\left(x+2y\right). Multiply \frac{x-2y}{x+2y} times \frac{x-2y}{x-2y}. Multiply \frac{x+2y}{x-2y} times \frac{x+2y}{x+2y}.

$$\frac{\left(\frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\right)\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Since \frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)} and \frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Do the multiplications in \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right).

$$\frac{\frac{x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Combine like terms in x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4xy}{4xy}.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(\frac{4xy}{4xy}+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Since \frac{4xy}{4xy} and \frac{x^{2}+4y^{2}}{4xy} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\times \frac{4xy+x^{2}+4y^{2}}{4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Multiply \frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} times \frac{4xy+x^{2}+4y^{2}}{4xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Express \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) as a single fraction.

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}}$$

Divide \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} by \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} by multiplying \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} by the reciprocal of \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}.

$$\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)\times 2xy}{\left(x-2y\right)\left(x+2y\right)\times 4xy\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

Cancel out 2xy in both numerator and denominator.

$$\frac{\left(2x^{2}+8y^{2}\right)\left(x^{2}+4xy+4y^{2}\right)}{2\left(x-2y\right)\left(x+2y\right)\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}{2x\left(x-2y\right)\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}$$

Cancel out 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right) in both numerator and denominator.

$$\frac{1}{x\left(x-2y\right)}$$

Expand the expression.

$$\frac{1}{x^{2}-2xy}$$

Expand

$$\frac{1}{x\left(x-2y\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2y and x-2y is \left(x-2y\right)\left(x+2y\right). Multiply \frac{x-2y}{x+2y} times \frac{x-2y}{x-2y}. Multiply \frac{x+2y}{x-2y} times \frac{x+2y}{x+2y}.

$$\frac{\left(\frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)}+\frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\right)\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Since \frac{\left(x-2y\right)\left(x-2y\right)}{\left(x-2y\right)\left(x+2y\right)} and \frac{\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)} have the same denominator, add them by adding their numerators.

$$\frac{\frac{\left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right)}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Do the multiplications in \left(x-2y\right)\left(x-2y\right)+\left(x+2y\right)\left(x+2y\right).

$$\frac{\frac{x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Combine like terms in x^{2}-2xy-2xy+4y^{2}+x^{2}+2xy+2xy+4y^{2}.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(1+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4xy}{4xy}.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\left(\frac{4xy}{4xy}+\frac{x^{2}+4y^{2}}{4xy}\right)}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Since \frac{4xy}{4xy} and \frac{x^{2}+4y^{2}}{4xy} have the same denominator, add them by adding their numerators.

$$\frac{\frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)}\times \frac{4xy+x^{2}+4y^{2}}{4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Multiply \frac{2x^{2}+8y^{2}}{\left(x-2y\right)\left(x+2y\right)} times \frac{4xy+x^{2}+4y^{2}}{4xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right)}$$

Express \frac{x^{2}+4y^{2}}{2xy}\left(x^{2}+2xy\right) as a single fraction.

$$\frac{\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy}}{\frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}}$$

Divide \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} by \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy} by multiplying \frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)}{\left(x-2y\right)\left(x+2y\right)\times 4xy} by the reciprocal of \frac{\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}{2xy}.

$$\frac{\left(2x^{2}+8y^{2}\right)\left(4xy+x^{2}+4y^{2}\right)\times 2xy}{\left(x-2y\right)\left(x+2y\right)\times 4xy\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

Cancel out 2xy in both numerator and denominator.

$$\frac{\left(2x^{2}+8y^{2}\right)\left(x^{2}+4xy+4y^{2}\right)}{2\left(x-2y\right)\left(x+2y\right)\left(x^{2}+4y^{2}\right)\left(x^{2}+2xy\right)}$$

Factor the expressions that are not already factored.

$$\frac{2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}{2x\left(x-2y\right)\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right)}$$

Cancel out 2\left(x+2y\right)^{2}\left(x^{2}+4y^{2}\right) in both numerator and denominator.

$$\frac{1}{x\left(x-2y\right)}$$

Expand the expression.

$$\frac{1}{x^{2}-2xy}$$