Solve (a^2+3)^3-(a^2-a)^3=

$$( a ^ { 2 } + 3 ) ^ { 3 } - ( a ^ { 2 } - a ) ^ { 3 } =$$

Evaluate

$$3a^{5}+6a^{4}+a^{3}+27a^{2}+27$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

To find the opposite of a^{6}-3a^{5}+3a^{4}-a^{3}, find the opposite of each term.

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

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

$$9a^{4}+27a^{2}+27+3a^{5}-3a^{4}+a^{3}$$

Combine 9a^{4} and -3a^{4} to get 6a^{4}.

$$6a^{4}+27a^{2}+27+3a^{5}+a^{3}$$

Expand

$$3a^{5}+6a^{4}+a^{3}+27a^{2}+27$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

To find the opposite of a^{6}-3a^{5}+3a^{4}-a^{3}, find the opposite of each term.

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

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

$$9a^{4}+27a^{2}+27+3a^{5}-3a^{4}+a^{3}$$

Combine 9a^{4} and -3a^{4} to get 6a^{4}.

$$6a^{4}+27a^{2}+27+3a^{5}+a^{3}$$