Solve (a^{2}-2a)^{3}+a(2a^{2}+3a)^2-2a^3(2a-1/2)^2-a^4(a+2)(a-12)

$$( a ^ { 2 } - 2 a ) ^ { 3 } + a ( 2 a ^ { 2 } + 3 a ) ^ { 2 } - 2 a ^ { 3 } ( 2 a - \frac { 1 } { 2 } ) ^ { 2 } - a ^ { 4 } ( a + 2 ) ( a - 12 )$$

Evaluate

$$\frac{\left(104a+1\right)a^{3}}{2}$$

Solution Steps

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

$$\left(a^{2}\right)^{3}-6\left(a^{2}\right)^{2}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6\left(a^{2}\right)^{2}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{4}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4\left(a^{2}\right)^{2}+12a^{2}a+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4a^{4}+12a^{2}a+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4a^{4}+12a^{3}+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Use the distributive property to multiply a by 4a^{4}+12a^{3}+9a^{2}.

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+4a^{5}+12a^{4}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine -6a^{5} and 4a^{5} to get -2a^{5}.

$$a^{6}-2a^{5}+12a^{4}-8a^{3}+12a^{4}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine 12a^{4} and 12a^{4} to get 24a^{4}.

$$a^{6}-2a^{5}+24a^{4}-8a^{3}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine -8a^{3} and 9a^{3} to get a^{3}.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-a^{4}\left(a+2\right)\left(a-12\right)$$

Use the distributive property to multiply a^{4} by a+2.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-\left(a^{5}+2a^{4}\right)\left(a-12\right)$$

Use the distributive property to multiply a^{5}+2a^{4} by a-12 and combine like terms.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-\left(a^{6}-10a^{5}-24a^{4}\right)$$

To find the opposite of a^{6}-10a^{5}-24a^{4}, find the opposite of each term.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-a^{6}+10a^{5}+24a^{4}$$

Use the distributive property to multiply -2a^{3} by 4a^{2}-2a+\frac{1}{4}.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-8a^{5}+4a^{4}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine -2a^{5} and -8a^{5} to get -10a^{5}.

$$a^{6}-10a^{5}+24a^{4}+a^{3}+4a^{4}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine 24a^{4} and 4a^{4} to get 28a^{4}.

$$a^{6}-10a^{5}+28a^{4}+a^{3}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine a^{3} and -\frac{1}{2}a^{3} to get \frac{1}{2}a^{3}.

$$a^{6}-10a^{5}+28a^{4}+\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine a^{6} and -a^{6} to get 0.

$$-10a^{5}+28a^{4}+\frac{1}{2}a^{3}+10a^{5}+24a^{4}$$

Combine -10a^{5} and 10a^{5} to get 0.

$$28a^{4}+\frac{1}{2}a^{3}+24a^{4}$$

Combine 28a^{4} and 24a^{4} to get 52a^{4}.

$$52a^{4}+\frac{1}{2}a^{3}$$

Expand

$$52a^{4}+\frac{a^{3}}{2}$$

Solution Steps

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

$$\left(a^{2}\right)^{3}-6\left(a^{2}\right)^{2}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6\left(a^{2}\right)^{2}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{4}a+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{2}a^{2}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(2a^{2}+3a\right)^{2}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4\left(a^{2}\right)^{2}+12a^{2}a+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4a^{4}+12a^{2}a+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+a\left(4a^{4}+12a^{3}+9a^{2}\right)-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Use the distributive property to multiply a by 4a^{4}+12a^{3}+9a^{2}.

$$a^{6}-6a^{5}+12a^{4}-8a^{3}+4a^{5}+12a^{4}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine -6a^{5} and 4a^{5} to get -2a^{5}.

$$a^{6}-2a^{5}+12a^{4}-8a^{3}+12a^{4}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine 12a^{4} and 12a^{4} to get 24a^{4}.

$$a^{6}-2a^{5}+24a^{4}-8a^{3}+9a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

Combine -8a^{3} and 9a^{3} to get a^{3}.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(2a-\frac{1}{2}\right)^{2}-a^{4}\left(a+2\right)\left(a-12\right)$$

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

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-a^{4}\left(a+2\right)\left(a-12\right)$$

Use the distributive property to multiply a^{4} by a+2.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-\left(a^{5}+2a^{4}\right)\left(a-12\right)$$

Use the distributive property to multiply a^{5}+2a^{4} by a-12 and combine like terms.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-\left(a^{6}-10a^{5}-24a^{4}\right)$$

To find the opposite of a^{6}-10a^{5}-24a^{4}, find the opposite of each term.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-2a^{3}\left(4a^{2}-2a+\frac{1}{4}\right)-a^{6}+10a^{5}+24a^{4}$$

Use the distributive property to multiply -2a^{3} by 4a^{2}-2a+\frac{1}{4}.

$$a^{6}-2a^{5}+24a^{4}+a^{3}-8a^{5}+4a^{4}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine -2a^{5} and -8a^{5} to get -10a^{5}.

$$a^{6}-10a^{5}+24a^{4}+a^{3}+4a^{4}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine 24a^{4} and 4a^{4} to get 28a^{4}.

$$a^{6}-10a^{5}+28a^{4}+a^{3}-\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine a^{3} and -\frac{1}{2}a^{3} to get \frac{1}{2}a^{3}.

$$a^{6}-10a^{5}+28a^{4}+\frac{1}{2}a^{3}-a^{6}+10a^{5}+24a^{4}$$

Combine a^{6} and -a^{6} to get 0.

$$-10a^{5}+28a^{4}+\frac{1}{2}a^{3}+10a^{5}+24a^{4}$$

Combine -10a^{5} and 10a^{5} to get 0.

$$28a^{4}+\frac{1}{2}a^{3}+24a^{4}$$

Combine 28a^{4} and 24a^{4} to get 52a^{4}.

$$52a^{4}+\frac{1}{2}a^{3}$$