Solve 1/(2n-1)^2+1/4n^2

$$\frac { 1 } { ( 2 n - 1 ) ^ { 2 } } + \frac { 1 } { 4 n ^ { 2 } }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2n-1\right)^{2} and 4n^{2} is 4n^{2}\left(2n-1\right)^{2}. Multiply \frac{1}{\left(2n-1\right)^{2}} times \frac{4n^{2}}{4n^{2}}. Multiply \frac{1}{4n^{2}} times \frac{\left(2n-1\right)^{2}}{\left(2n-1\right)^{2}}.

$$\frac{4n^{2}}{4n^{2}\left(2n-1\right)^{2}}+\frac{\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}}$$

Since \frac{4n^{2}}{4n^{2}\left(2n-1\right)^{2}} and \frac{\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{4n^{2}+\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}}$$

Do the multiplications in 4n^{2}+\left(2n-1\right)^{2}.

$$\frac{4n^{2}+4n^{2}-4n+1}{4n^{2}\left(2n-1\right)^{2}}$$

Combine like terms in 4n^{2}+4n^{2}-4n+1.

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

Expand 4n^{2}\left(2n-1\right)^{2}.

$$\frac{8n^{2}-4n+1}{16n^{4}-16n^{3}+4n^{2}}$$

Expand

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(2n-1\right)^{2} and 4n^{2} is 4n^{2}\left(2n-1\right)^{2}. Multiply \frac{1}{\left(2n-1\right)^{2}} times \frac{4n^{2}}{4n^{2}}. Multiply \frac{1}{4n^{2}} times \frac{\left(2n-1\right)^{2}}{\left(2n-1\right)^{2}}.

$$\frac{4n^{2}}{4n^{2}\left(2n-1\right)^{2}}+\frac{\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}}$$

Since \frac{4n^{2}}{4n^{2}\left(2n-1\right)^{2}} and \frac{\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{4n^{2}+\left(2n-1\right)^{2}}{4n^{2}\left(2n-1\right)^{2}}$$

Do the multiplications in 4n^{2}+\left(2n-1\right)^{2}.

$$\frac{4n^{2}+4n^{2}-4n+1}{4n^{2}\left(2n-1\right)^{2}}$$

Combine like terms in 4n^{2}+4n^{2}-4n+1.

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

Expand 4n^{2}\left(2n-1\right)^{2}.

$$\frac{8n^{2}-4n+1}{16n^{4}-16n^{3}+4n^{2}}$$