Solve (1/2a^2-3/2a)^3

$$( \frac { 1 } { 2 } a ^ { 2 } - \frac { 3 } { 2 } a ) ^ { 3 }$$

Evaluate

$$\frac{\left(a\left(a-3\right)\right)^{3}}{8}$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

Expand

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

Solution Steps

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

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

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

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

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

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

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

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

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

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