 QUESTION

# What is the value of ${(a + b)^3} - {(a - b)^3}$ ?(a). ${b^3} + 3{a^2}b$(b). $2({b^3} + 3{a^2}b)$(c). $2({a^3} + 3a{b^2})$(d). 0

Hint: Recall the formula for ${(a + b)^3}$ and ${(a - b)^3}$, which is ${(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}$ , and ${(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}$ respectively. Use these formulae to evaluate ${(a + b)^3} - {(a - b)^3}$.

Complete Step-by-Step solution:
In this problem, we need to evaluate ${(a + b)^3} - {(a - b)^3}$. For that first, we find the expansion of ${(a + b)^3}$ and ${(a - b)^3}$ and then evaluate them.
First, let us evaluate ${(a + b)^3}$. We know that ${(a + b)^3}$ can be written as $(a + b)(a + b)(a + b)$. Hence, we have:
$\Rightarrow$ ${(a + b)^3} = (a + b)(a + b)(a + b)$
We multiply the first two terms which is nothing but the expression for ${(a + b)^2}$.
$\Rightarrow$ ${(a + b)^3} = (a.a + a.b + b.a + b.b)(a + b)$
Simplifying the above equation, we get as follows:
$\Rightarrow$ ${(a + b)^3} = ({a^2} + 2ab + {b^2})(a + b)$
Now, we multiply the remaining terms to get the final expression. Hence, we have:
$\Rightarrow$ ${(a + b)^3} = {a^2}.a + 2ab.a + {b^2}.a + {a^2}.b + 2ab.b + {b^2}.b$
Simplifying the above equation, we get:
$\Rightarrow$ ${(a + b)^3} = {a^3} + 2{a^2}b + a{b^2} + {a^2}b + 2a{b^2} + {b^3}$
Adding the common terms together, we have:
$\Rightarrow$ ${(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}.............(1)$
Now, let us evaluate ${(a - b)^3}$. We know that ${(a - b)^3}$ can be written as $(a - b)(a - b)(a - b)$. Hence, we have:
$\Rightarrow$ ${(a - b)^3} = (a - b)(a - b)(a - b)$
We multiply the first two terms which is nothing but the expression for ${(a - b)^2}$.
$\Rightarrow$ ${(a - b)^3} = (a.a - a.b - b.a + b.b)(a - b)$
Simplifying the above equation, we get as follows:
$\Rightarrow$ ${(a - b)^3} = ({a^2} - 2ab + {b^2})(a - b)$
Now, we multiply the remaining terms to get the final expression. Hence, we have:
$\Rightarrow$ ${(a - b)^3} = {a^2}.a - 2ab.a + {b^2}.a - {a^2}.b + 2ab.b - {b^2}.b$
Simplifying the above equation, we get:
$\Rightarrow$ ${(a - b)^3} = {a^3} - 2{a^2}b + a{b^2} - {a^2}b + 2a{b^2} - {b^3}$
Adding the common terms together, we have:
$\Rightarrow$ ${(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}.............(2)$
Now, we subtract equation (2) from equation (1) to obtain the value of ${(a + b)^3} - {(a - b)^3}$. Hence, we have as follows:
$\Rightarrow$ ${(a + b)^3} - {(a - b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} - ({a^3} - 3{a^2}b + 3a{b^2} - {b^3})$
$\Rightarrow$ ${(a + b)^3} - {(a - b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3} - {a^3} + 3{a^2}b - 3a{b^2} + {b^3}$
$\Rightarrow$ ${(a + b)^3} - {(a - b)^3} = 3a{b^2} + {b^3} + 3{a^2}b + {b^3}$
$\Rightarrow$ ${(a + b)^3} - {(a - b)^3} = 2{b^3} + 6{a^2}b$
$\Rightarrow$ ${(a + b)^3} - {(a - b)^3} = 2({b^3} + 3{a^2}b)$