Solve 1/y-y+8/16-y^2-2/y-4

$$\frac { 1 } { y } - \frac { y + 8 } { 16 - y ^ { 2 } } - \frac { 2 } { y - 4 }$$

Evaluate

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

Short Solution Steps

Factor 16-y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and \left(y-4\right)\left(-y-4\right) is y\left(y-4\right)\left(-y-4\right). Multiply \frac{1}{y} times \frac{\left(y-4\right)\left(-y-4\right)}{\left(y-4\right)\left(-y-4\right)}. Multiply \frac{y+8}{\left(y-4\right)\left(-y-4\right)} times \frac{y}{y}.

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

Since \frac{\left(y-4\right)\left(-y-4\right)}{y\left(y-4\right)\left(-y-4\right)} and \frac{\left(y+8\right)y}{y\left(y-4\right)\left(-y-4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(y-4\right)\left(-y-4\right)-\left(y+8\right)y.

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

Combine like terms in -y^{2}-4y+4y+16-y^{2}-8y.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y\left(y-4\right)\left(-y-4\right) and y-4 is y\left(y-4\right)\left(-y-4\right). Multiply \frac{2}{y-4} times \frac{y\left(-y-4\right)}{y\left(-y-4\right)}.

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

Since \frac{-2y^{2}-8y+16}{y\left(y-4\right)\left(-y-4\right)} and \frac{2y\left(-y-4\right)}{y\left(y-4\right)\left(-y-4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -2y^{2}-8y+16-2y\left(-y-4\right).

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

Combine like terms in -2y^{2}-8y+16+2y^{2}+8y.

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

Expand y\left(y-4\right)\left(-y-4\right).

$$\frac{16}{-y^{3}+16y}$$

Expand

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

Short Solution Steps

Factor 16-y^{2}.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y and \left(y-4\right)\left(-y-4\right) is y\left(y-4\right)\left(-y-4\right). Multiply \frac{1}{y} times \frac{\left(y-4\right)\left(-y-4\right)}{\left(y-4\right)\left(-y-4\right)}. Multiply \frac{y+8}{\left(y-4\right)\left(-y-4\right)} times \frac{y}{y}.

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

Since \frac{\left(y-4\right)\left(-y-4\right)}{y\left(y-4\right)\left(-y-4\right)} and \frac{\left(y+8\right)y}{y\left(y-4\right)\left(-y-4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(y-4\right)\left(-y-4\right)-\left(y+8\right)y.

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

Combine like terms in -y^{2}-4y+4y+16-y^{2}-8y.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of y\left(y-4\right)\left(-y-4\right) and y-4 is y\left(y-4\right)\left(-y-4\right). Multiply \frac{2}{y-4} times \frac{y\left(-y-4\right)}{y\left(-y-4\right)}.

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

Since \frac{-2y^{2}-8y+16}{y\left(y-4\right)\left(-y-4\right)} and \frac{2y\left(-y-4\right)}{y\left(y-4\right)\left(-y-4\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in -2y^{2}-8y+16-2y\left(-y-4\right).

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

Combine like terms in -2y^{2}-8y+16+2y^{2}+8y.

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

Expand y\left(y-4\right)\left(-y-4\right).

$$\frac{16}{-y^{3}+16y}$$