Solve y+2/y-2-y-2/y+2-16/y^2-4

$$\frac { y + 2 } { y - 2 } - \frac { y - 2 } { y + 2 } - \frac { 16 } { y ^ { 2 } - 4 }$$

Evaluate

$$\frac{8}{y+2}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{8y}{\left(y-2\right)\left(y+2\right)}-\frac{16}{y^{2}-4}$$

Factor y^{2}-4.

$$\frac{8y}{\left(y-2\right)\left(y+2\right)}-\frac{16}{\left(y-2\right)\left(y+2\right)}$$

Since \frac{8y}{\left(y-2\right)\left(y+2\right)} and \frac{16}{\left(y-2\right)\left(y+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8y-16}{\left(y-2\right)\left(y+2\right)}$$

Factor the expressions that are not already factored in \frac{8y-16}{\left(y-2\right)\left(y+2\right)}.

$$\frac{8\left(y-2\right)}{\left(y-2\right)\left(y+2\right)}$$

Cancel out y-2 in both numerator and denominator.

$$\frac{8}{y+2}$$

Expand

$$\frac{8}{y+2}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{8y}{\left(y-2\right)\left(y+2\right)}-\frac{16}{y^{2}-4}$$

Factor y^{2}-4.

$$\frac{8y}{\left(y-2\right)\left(y+2\right)}-\frac{16}{\left(y-2\right)\left(y+2\right)}$$

Since \frac{8y}{\left(y-2\right)\left(y+2\right)} and \frac{16}{\left(y-2\right)\left(y+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{8y-16}{\left(y-2\right)\left(y+2\right)}$$

Factor the expressions that are not already factored in \frac{8y-16}{\left(y-2\right)\left(y+2\right)}.

$$\frac{8\left(y-2\right)}{\left(y-2\right)\left(y+2\right)}$$

Cancel out y-2 in both numerator and denominator.

$$\frac{8}{y+2}$$