Solve 324/n^3(n(2n+1)(n+1)/6)

$$\frac { 324 } { n ^ { 3 } } ( \frac { n ( 2 n + 1 ) ( n + 1 ) } { 6 } )$$

Evaluate

$$108+\frac{162}{n}+\frac{54}{n^{2}}$$

Short Solution Steps

Use the distributive property to multiply n by 2n+1.

$$\frac{324}{n^{3}}\times \frac{\left(2n^{2}+n\right)\left(n+1\right)}{6}$$

Use the distributive property to multiply 2n^{2}+n by n+1 and combine like terms.

$$\frac{324}{n^{3}}\times \frac{2n^{3}+3n^{2}+n}{6}$$

Multiply \frac{324}{n^{3}} times \frac{2n^{3}+3n^{2}+n}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{324\left(2n^{3}+3n^{2}+n\right)}{n^{3}\times 6}$$

Cancel out 6 in both numerator and denominator.

$$\frac{54\left(2n^{3}+3n^{2}+n\right)}{n^{3}}$$

Factor the expressions that are not already factored.

$$\frac{54n\left(n+1\right)\left(2n+1\right)}{n^{3}}$$

Cancel out n in both numerator and denominator.

$$\frac{54\left(n+1\right)\left(2n+1\right)}{n^{2}}$$

Expand the expression.

$$\frac{108n^{2}+162n+54}{n^{2}}$$

Expand

$$108+\frac{162}{n}+\frac{54}{n^{2}}$$

Short Solution Steps

Use the distributive property to multiply n by 2n+1.

$$\frac{324}{n^{3}}\times \frac{\left(2n^{2}+n\right)\left(n+1\right)}{6}$$

Use the distributive property to multiply 2n^{2}+n by n+1 and combine like terms.

$$\frac{324}{n^{3}}\times \frac{2n^{3}+3n^{2}+n}{6}$$

Multiply \frac{324}{n^{3}} times \frac{2n^{3}+3n^{2}+n}{6} by multiplying numerator times numerator and denominator times denominator.

$$\frac{324\left(2n^{3}+3n^{2}+n\right)}{n^{3}\times 6}$$

Cancel out 6 in both numerator and denominator.

$$\frac{54\left(2n^{3}+3n^{2}+n\right)}{n^{3}}$$

Factor the expressions that are not already factored.

$$\frac{54n\left(n+1\right)\left(2n+1\right)}{n^{3}}$$

Cancel out n in both numerator and denominator.

$$\frac{54\left(n+1\right)\left(2n+1\right)}{n^{2}}$$

Expand the expression.

$$\frac{108n^{2}+162n+54}{n^{2}}$$