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

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

Expand

$$13a^{3}+90a^{2}+72a+78$$

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}.

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

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

$$16a^{3}+72a^{2}+108a+54-3\left(a-2\right)^{3}$$

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

$$16a^{3}+72a^{2}+108a+54-3\left(a^{3}-6a^{2}+12a-8\right)$$

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

$$16a^{3}+72a^{2}+108a+54-3a^{3}+18a^{2}-36a+24$$

Combine 16a^{3} and -3a^{3} to get 13a^{3}.

$$13a^{3}+72a^{2}+108a+54+18a^{2}-36a+24$$

Combine 72a^{2} and 18a^{2} to get 90a^{2}.

$$13a^{3}+90a^{2}+108a+54-36a+24$$

Combine 108a and -36a to get 72a.

$$13a^{3}+90a^{2}+72a+54+24$$

Add 54 and 24 to get 78.

$$13a^{3}+90a^{2}+72a+78$$

Evaluate

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

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}.

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

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

$$16a^{3}+72a^{2}+108a+54-3\left(a-2\right)^{3}$$

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

$$16a^{3}+72a^{2}+108a+54-3\left(a^{3}-6a^{2}+12a-8\right)$$

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

$$16a^{3}+72a^{2}+108a+54-3a^{3}+18a^{2}-36a+24$$

Combine 16a^{3} and -3a^{3} to get 13a^{3}.

$$13a^{3}+72a^{2}+108a+54+18a^{2}-36a+24$$

Combine 72a^{2} and 18a^{2} to get 90a^{2}.

$$13a^{3}+90a^{2}+108a+54-36a+24$$

Combine 108a and -36a to get 72a.

$$13a^{3}+90a^{2}+72a+54+24$$

Add 54 and 24 to get 78.

$$13a^{3}+90a^{2}+72a+78$$