Solve (x+4y/x^{2-4xy}-x-4y/x^2+4xy):4y^2/x^2-16y^2

$$( \frac { x + 4 y } { x ^ { 2 } - 4 x y } - \frac { x - 4 y } { x ^ { 2 } + 4 x y } ) : \frac { 4 y ^ { 2 } } { x ^ { 2 } - 16 y ^ { 2 } }$$

Evaluate

$$\frac{4}{y}$$

Short Solution Steps

Factor x^{2}-4xy. Factor x^{2}+4xy.

$$\frac{\frac{x+4y}{x\left(x-4y\right)}-\frac{x-4y}{x\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-4y\right) and x\left(x+4y\right) is x\left(x-4y\right)\left(x+4y\right). Multiply \frac{x+4y}{x\left(x-4y\right)} times \frac{x+4y}{x+4y}. Multiply \frac{x-4y}{x\left(x+4y\right)} times \frac{x-4y}{x-4y}.

$$\frac{\frac{\left(x+4y\right)\left(x+4y\right)}{x\left(x-4y\right)\left(x+4y\right)}-\frac{\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Since \frac{\left(x+4y\right)\left(x+4y\right)}{x\left(x-4y\right)\left(x+4y\right)} and \frac{\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(x+4y\right)\left(x+4y\right)-\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Do the multiplications in \left(x+4y\right)\left(x+4y\right)-\left(x-4y\right)\left(x-4y\right).

$$\frac{\frac{x^{2}+4xy+4xy+16y^{2}-x^{2}+4xy+4xy-16y^{2}}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Combine like terms in x^{2}+4xy+4xy+16y^{2}-x^{2}+4xy+4xy-16y^{2}.

$$\frac{\frac{16xy}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{16y}{\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Divide \frac{16y}{\left(x-4y\right)\left(x+4y\right)} by \frac{4y^{2}}{x^{2}-16y^{2}} by multiplying \frac{16y}{\left(x-4y\right)\left(x+4y\right)} by the reciprocal of \frac{4y^{2}}{x^{2}-16y^{2}}.

$$\frac{16y\left(x^{2}-16y^{2}\right)}{\left(x-4y\right)\left(x+4y\right)\times 4y^{2}}$$

Cancel out 4y in both numerator and denominator.

$$\frac{4\left(x^{2}-16y^{2}\right)}{y\left(x-4y\right)\left(x+4y\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-4y\right)\left(x+4y\right) in both numerator and denominator.

$$\frac{4}{y}$$

Expand

$$\frac{4}{y}$$

Short Solution Steps

Factor x^{2}-4xy. Factor x^{2}+4xy.

$$\frac{\frac{x+4y}{x\left(x-4y\right)}-\frac{x-4y}{x\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x\left(x-4y\right) and x\left(x+4y\right) is x\left(x-4y\right)\left(x+4y\right). Multiply \frac{x+4y}{x\left(x-4y\right)} times \frac{x+4y}{x+4y}. Multiply \frac{x-4y}{x\left(x+4y\right)} times \frac{x-4y}{x-4y}.

$$\frac{\frac{\left(x+4y\right)\left(x+4y\right)}{x\left(x-4y\right)\left(x+4y\right)}-\frac{\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Since \frac{\left(x+4y\right)\left(x+4y\right)}{x\left(x-4y\right)\left(x+4y\right)} and \frac{\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(x+4y\right)\left(x+4y\right)-\left(x-4y\right)\left(x-4y\right)}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Do the multiplications in \left(x+4y\right)\left(x+4y\right)-\left(x-4y\right)\left(x-4y\right).

$$\frac{\frac{x^{2}+4xy+4xy+16y^{2}-x^{2}+4xy+4xy-16y^{2}}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Combine like terms in x^{2}+4xy+4xy+16y^{2}-x^{2}+4xy+4xy-16y^{2}.

$$\frac{\frac{16xy}{x\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Cancel out x in both numerator and denominator.

$$\frac{\frac{16y}{\left(x-4y\right)\left(x+4y\right)}}{\frac{4y^{2}}{x^{2}-16y^{2}}}$$

Divide \frac{16y}{\left(x-4y\right)\left(x+4y\right)} by \frac{4y^{2}}{x^{2}-16y^{2}} by multiplying \frac{16y}{\left(x-4y\right)\left(x+4y\right)} by the reciprocal of \frac{4y^{2}}{x^{2}-16y^{2}}.

$$\frac{16y\left(x^{2}-16y^{2}\right)}{\left(x-4y\right)\left(x+4y\right)\times 4y^{2}}$$

Cancel out 4y in both numerator and denominator.

$$\frac{4\left(x^{2}-16y^{2}\right)}{y\left(x-4y\right)\left(x+4y\right)}$$

Factor the expressions that are not already factored.

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

Cancel out \left(x-4y\right)\left(x+4y\right) in both numerator and denominator.

$$\frac{4}{y}$$