Solve (a^{3}+1/27+a^2+1/3)*a-2a*(a^2+1/6)+(-3/5a-frac{3}{2}a^4+frac{1}{5}a^3):(+frac{9}{5}a)

$$( a ^ { 3 } + \frac { 1 } { 27 } + a ^ { 2 } + \frac { 1 } { 3 } ) \cdot a - 2 a \cdot ( a ^ { 2 } + \frac { 1 } { 6 } ) + ( - \frac { 3 } { 5 } a - \frac { 3 } { 2 } a ^ { 4 } + \frac { 1 } { 5 } a ^ { 3 } ) : ( + \frac { 9 } { 5 } a )$$

Evaluate

$$a^{4}-\frac{11a^{3}}{6}+\frac{a^{2}}{9}+\frac{a}{27}-\frac{1}{3}$$

Short Solution Steps

Add \frac{1}{27} and \frac{1}{3} to get \frac{10}{27}.

$$\left(a^{3}+\frac{10}{27}+a^{2}\right)a-2a\left(a^{2}+\frac{1}{6}\right)+\frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}$$

Use the distributive property to multiply a^{3}+\frac{10}{27}+a^{2} by a.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}$$

Factor the expressions that are not already factored in \frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{\frac{1}{10}a\left(-15a^{3}+2a^{2}-6\right)}{\frac{9}{5}a}$$

Cancel out a in both numerator and denominator.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{\frac{1}{10}\left(-15a^{3}+2a^{2}-6\right)}{\frac{9}{5}}$$

Divide \frac{1}{10}\left(-15a^{3}+2a^{2}-6\right) by \frac{9}{5} to get \frac{1}{18}\left(-15a^{3}+2a^{2}-6\right).

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{1}{18}\left(-15a^{3}+2a^{2}-6\right)$$

Use the distributive property to multiply \frac{1}{18} by -15a^{3}+2a^{2}-6.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Use the distributive property to multiply -2a by a^{2}+\frac{1}{6}.

$$a^{4}+\frac{10}{27}a+a^{3}-2a^{3}-\frac{1}{3}a-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine a^{3} and -2a^{3} to get -a^{3}.

$$a^{4}+\frac{10}{27}a-a^{3}-\frac{1}{3}a-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine \frac{10}{27}a and -\frac{1}{3}a to get \frac{1}{27}a.

$$a^{4}+\frac{1}{27}a-a^{3}-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine -a^{3} and -\frac{5}{6}a^{3} to get -\frac{11}{6}a^{3}.

$$a^{4}+\frac{1}{27}a-\frac{11}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Expand

$$a^{4}-\frac{11a^{3}}{6}+\frac{a^{2}}{9}+\frac{a}{27}-\frac{1}{3}$$

Short Solution Steps

Add \frac{1}{27} and \frac{1}{3} to get \frac{10}{27}.

$$\left(a^{3}+\frac{10}{27}+a^{2}\right)a-2a\left(a^{2}+\frac{1}{6}\right)+\frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}$$

Use the distributive property to multiply a^{3}+\frac{10}{27}+a^{2} by a.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}$$

Factor the expressions that are not already factored in \frac{-\frac{3}{5}a-\frac{3}{2}a^{4}+\frac{1}{5}a^{3}}{\frac{9}{5}a}.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{\frac{1}{10}a\left(-15a^{3}+2a^{2}-6\right)}{\frac{9}{5}a}$$

Cancel out a in both numerator and denominator.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{\frac{1}{10}\left(-15a^{3}+2a^{2}-6\right)}{\frac{9}{5}}$$

Divide \frac{1}{10}\left(-15a^{3}+2a^{2}-6\right) by \frac{9}{5} to get \frac{1}{18}\left(-15a^{3}+2a^{2}-6\right).

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)+\frac{1}{18}\left(-15a^{3}+2a^{2}-6\right)$$

Use the distributive property to multiply \frac{1}{18} by -15a^{3}+2a^{2}-6.

$$a^{4}+\frac{10}{27}a+a^{3}-2a\left(a^{2}+\frac{1}{6}\right)-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Use the distributive property to multiply -2a by a^{2}+\frac{1}{6}.

$$a^{4}+\frac{10}{27}a+a^{3}-2a^{3}-\frac{1}{3}a-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine a^{3} and -2a^{3} to get -a^{3}.

$$a^{4}+\frac{10}{27}a-a^{3}-\frac{1}{3}a-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine \frac{10}{27}a and -\frac{1}{3}a to get \frac{1}{27}a.

$$a^{4}+\frac{1}{27}a-a^{3}-\frac{5}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$

Combine -a^{3} and -\frac{5}{6}a^{3} to get -\frac{11}{6}a^{3}.

$$a^{4}+\frac{1}{27}a-\frac{11}{6}a^{3}+\frac{1}{9}a^{2}-\frac{1}{3}$$