Solve (1-2/3a^{3})^{3}+(2a+1/2a^{2})^3-1/3(-a)^4(3/4-frac{8}{9}a^5)-frac{4}{3}a^6

$$( 1 - \frac { 2 } { 3 } a ^ { 3 } ) ^ { 3 } + ( 2 a + \frac { 1 } { 2 } a ^ { 2 } ) ^ { 3 } - \frac { 1 } { 3 } ( - a ) ^ { 4 } ( \frac { 3 } { 4 } - \frac { 8 } { 9 } a ^ { 5 } ) - \frac { 4 } { 3 } a ^ { 6 }$$

Evaluate

$$\frac{a^{6}}{8}+\frac{3a^{5}}{2}+\frac{23a^{4}}{4}+6a^{3}+1$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(1-\frac{2}{3}a^{3}\right)^{3}.

$$1-2a^{3}+\frac{4}{3}\left(a^{3}\right)^{2}-\frac{8}{27}\left(a^{3}\right)^{3}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}\left(a^{3}\right)^{3}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(2a+\frac{1}{2}a^{2}\right)^{3}.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{2}a^{2}+\frac{3}{2}a\left(a^{2}\right)^{2}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a\left(a^{2}\right)^{2}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}aa^{4}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}a^{6}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Combine -2a^{3} and 8a^{3} to get 6a^{3}.

$$1+6a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}a^{6}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Combine \frac{4}{3}a^{6} and \frac{1}{8}a^{6} to get \frac{35}{24}a^{6}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Expand \left(-a\right)^{4}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}\left(-1\right)^{4}a^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Calculate -1 to the power of 4 and get 1.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}a^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Use the distributive property to multiply -\frac{1}{3}a^{4} by \frac{3}{4}-\frac{8}{9}a^{5}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{4}a^{4}+\frac{8}{27}a^{9}-\frac{4}{3}a^{6}$$

Combine 6a^{4} and -\frac{1}{4}a^{4} to get \frac{23}{4}a^{4}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}+\frac{8}{27}a^{9}-\frac{4}{3}a^{6}$$

Combine -\frac{8}{27}a^{9} and \frac{8}{27}a^{9} to get 0.

$$1+6a^{3}+\frac{35}{24}a^{6}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}-\frac{4}{3}a^{6}$$

Combine \frac{35}{24}a^{6} and -\frac{4}{3}a^{6} to get \frac{1}{8}a^{6}.

$$1+6a^{3}+\frac{1}{8}a^{6}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}$$

Expand

$$\frac{a^{6}}{8}+\frac{3a^{5}}{2}+\frac{23a^{4}}{4}+6a^{3}+1$$

Solution Steps

Use binomial theorem \left(p-q\right)^{3}=p^{3}-3p^{2}q+3pq^{2}-q^{3} to expand \left(1-\frac{2}{3}a^{3}\right)^{3}.

$$1-2a^{3}+\frac{4}{3}\left(a^{3}\right)^{2}-\frac{8}{27}\left(a^{3}\right)^{3}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 2 to get 6.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}\left(a^{3}\right)^{3}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 3 and 3 to get 9.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+\left(2a+\frac{1}{2}a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Use binomial theorem \left(p+q\right)^{3}=p^{3}+3p^{2}q+3pq^{2}+q^{3} to expand \left(2a+\frac{1}{2}a^{2}\right)^{3}.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{2}a^{2}+\frac{3}{2}a\left(a^{2}\right)^{2}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a\left(a^{2}\right)^{2}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}aa^{4}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To multiply powers of the same base, add their exponents. Add 1 and 4 to get 5.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}\left(a^{2}\right)^{3}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 3 to get 6.

$$1-2a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+8a^{3}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}a^{6}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Combine -2a^{3} and 8a^{3} to get 6a^{3}.

$$1+6a^{3}+\frac{4}{3}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}+\frac{1}{8}a^{6}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Combine \frac{4}{3}a^{6} and \frac{1}{8}a^{6} to get \frac{35}{24}a^{6}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}\left(-a\right)^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Expand \left(-a\right)^{4}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}\left(-1\right)^{4}a^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Calculate -1 to the power of 4 and get 1.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{3}a^{4}\left(\frac{3}{4}-\frac{8}{9}a^{5}\right)-\frac{4}{3}a^{6}$$

Use the distributive property to multiply -\frac{1}{3}a^{4} by \frac{3}{4}-\frac{8}{9}a^{5}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+6a^{4}+\frac{3}{2}a^{5}-\frac{1}{4}a^{4}+\frac{8}{27}a^{9}-\frac{4}{3}a^{6}$$

Combine 6a^{4} and -\frac{1}{4}a^{4} to get \frac{23}{4}a^{4}.

$$1+6a^{3}+\frac{35}{24}a^{6}-\frac{8}{27}a^{9}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}+\frac{8}{27}a^{9}-\frac{4}{3}a^{6}$$

Combine -\frac{8}{27}a^{9} and \frac{8}{27}a^{9} to get 0.

$$1+6a^{3}+\frac{35}{24}a^{6}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}-\frac{4}{3}a^{6}$$

Combine \frac{35}{24}a^{6} and -\frac{4}{3}a^{6} to get \frac{1}{8}a^{6}.

$$1+6a^{3}+\frac{1}{8}a^{6}+\frac{23}{4}a^{4}+\frac{3}{2}a^{5}$$