Solve 8left(2a+3right)^3-27left(a-2right)^3

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

Evaluate

$$\left(a+12\right)\left(37a^{2}+6a+36\right)$$

Solution Steps

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

$$8\left(8a^{3}+36a^{2}+54a+27\right)-27\left(a-2\right)^{3}$$

Use the distributive property to multiply 8 by 8a^{3}+36a^{2}+54a+27.

$$64a^{3}+288a^{2}+432a+216-27\left(a-2\right)^{3}$$

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

$$64a^{3}+288a^{2}+432a+216-27\left(a^{3}-6a^{2}+12a-8\right)$$

Use the distributive property to multiply -27 by a^{3}-6a^{2}+12a-8.

$$64a^{3}+288a^{2}+432a+216-27a^{3}+162a^{2}-324a+216$$

Combine 64a^{3} and -27a^{3} to get 37a^{3}.

$$37a^{3}+288a^{2}+432a+216+162a^{2}-324a+216$$

Combine 288a^{2} and 162a^{2} to get 450a^{2}.

$$37a^{3}+450a^{2}+432a+216-324a+216$$

Combine 432a and -324a to get 108a.

$$37a^{3}+450a^{2}+108a+216+216$$

Add 216 and 216 to get 432.

$$37a^{3}+450a^{2}+108a+432$$

Expand

$$37a^{3}+450a^{2}+108a+432$$

Solution Steps

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

$$8\left(8a^{3}+36a^{2}+54a+27\right)-27\left(a-2\right)^{3}$$

Use the distributive property to multiply 8 by 8a^{3}+36a^{2}+54a+27.

$$64a^{3}+288a^{2}+432a+216-27\left(a-2\right)^{3}$$

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

$$64a^{3}+288a^{2}+432a+216-27\left(a^{3}-6a^{2}+12a-8\right)$$

Use the distributive property to multiply -27 by a^{3}-6a^{2}+12a-8.

$$64a^{3}+288a^{2}+432a+216-27a^{3}+162a^{2}-324a+216$$

Combine 64a^{3} and -27a^{3} to get 37a^{3}.

$$37a^{3}+288a^{2}+432a+216+162a^{2}-324a+216$$

Combine 288a^{2} and 162a^{2} to get 450a^{2}.

$$37a^{3}+450a^{2}+432a+216-324a+216$$

Combine 432a and -324a to get 108a.

$$37a^{3}+450a^{2}+108a+216+216$$

Add 216 and 216 to get 432.

$$37a^{3}+450a^{2}+108a+432$$