Solve {l}{1/(n+2)^2}{+1/n^2-4}

$$\left. \begin{array} { l } { \frac { 1 } { ( n + 2 ) ^ { 2 } } } \\ { + \frac { 1 } { n ^ { 2 } - 4 } } \end{array} \right.$$

Evaluate

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

Short Solution Steps

Factor n^{2}-4.

$$\frac{1}{\left(n+2\right)^{2}}+\frac{1}{\left(n-2\right)\left(n+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(n+2\right)^{2} and \left(n-2\right)\left(n+2\right) is \left(n-2\right)\left(n+2\right)^{2}. Multiply \frac{1}{\left(n+2\right)^{2}} times \frac{n-2}{n-2}. Multiply \frac{1}{\left(n-2\right)\left(n+2\right)} times \frac{n+2}{n+2}.

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

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

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

Combine like terms in n-2+n+2.

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

Expand \left(n-2\right)\left(n+2\right)^{2}.

$$\frac{2n}{n^{3}+2n^{2}-4n-8}$$

Expand

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

Short Solution Steps

Factor n^{2}-4.

$$\frac{1}{\left(n+2\right)^{2}}+\frac{1}{\left(n-2\right)\left(n+2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(n+2\right)^{2} and \left(n-2\right)\left(n+2\right) is \left(n-2\right)\left(n+2\right)^{2}. Multiply \frac{1}{\left(n+2\right)^{2}} times \frac{n-2}{n-2}. Multiply \frac{1}{\left(n-2\right)\left(n+2\right)} times \frac{n+2}{n+2}.

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

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

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

Combine like terms in n-2+n+2.

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

Expand \left(n-2\right)\left(n+2\right)^{2}.

$$\frac{2n}{n^{3}+2n^{2}-4n-8}$$