Question

Simplify: \[{(2a + b)^3} - {(2a - b)^3}\]

Answer 1

Hint: The given question requires us to solve a cubic equation. Cubic equations are those equations that have power of three. It can be simplified with the help of formula \[{(a + b)^3}\] and \[{(a - b)^3}\] . We will factorize the question after using the formula to arrive at the solution.

Complete step-by-step answer:
The given question is an algebraic equation with power of three called cubic equations. We have to identify the form of the cubic equation and then apply a suitable formula to arrive at the answer.
The equation can be rewritten in the form of \[{(a + b)^3}\] and \[{(a - b)^3}\] using the following formula:
 \[{(a + b)^3} = {a^3} + {b^3} + 3ab(a + b)\]
 \[{(a - b)^3} = {a^3} - {b^3} - 3ab(a - b)\]
Let us solve \[{(2a + b)^3}\] first by comparing the above formula. We will get:
 \[{(2a + b)^3} = {(2a)^3} + {b^3} + 3(2a)(b)(2a + b)\]
Multiplying the last term, we get,
 \[{(2a + b)^3} = 8{a^3} + {b^3} + 3{(2a)^2}(b) + 3(2a){(b)^2}\]
 \[{(2a + b)^3} = 8{a^3} + {b^3} + 3(4{a^2})(b) + 3(2a){(b)^2}\]
Opening up and solving the brackets, we get,
 \[{(2a + b)^3} = 8{a^3} + {b^3} + 12{a^2}b + 6a{b^2}\] ------(1)
Now we will solve \[{(2a - b)^3}\] by comparing with the formula. We will get:
 \[{(2a - b)^3} = {(2a)^3} - {b^3} - 3(2a)(b)(2a - b)\]
Multiplying the last term, we get,
 \[{(2a - b)^3} = 8{a^3} - {b^3} - 3{(2a)^2}(b) + 3(2a){(b)^2}\]
 \[{(2a - b)^3} = 8{a^3} - {b^3} - 3(4{a^2})(b) + 3(2a){(b)^2}\]
Opening up and solving the brackets, we get,
 \[{(2a - b)^3} = 8{a^3} - {b^3} - 12{a^2}b + 6a{b^2}\] -----(2)
Substituting the value of equation (1) and (2), in the equation given in the question, we will get,
 \[{(2a + b)^3} - {(2a - b)^3} = 8{a^3} + {b^3} + 12{a^2}b + 6a{b^2} - (8{a^3} - {b^3} - 12{a^2}b + 6a{b^2})\]
Opening up the brackets and reversing the sign, we will get,
 \[{(2a + b)^3} - {(2a - b)^3} = 8{a^3} + {b^3} + 12{a^2}b + 6a{b^2} - 8{a^3} + {b^3} + 12{a^2}b - 6a{b^2}\]
Writing common terms together for simplicity of addition and subtraction, we will get,
 \[{(2a + b)^3} - {(2a - b)^3} = 8{a^3} - 8{a^3} + {b^3} + {b^3} + 12{a^2}b + 12{a^2}b + 6a{b^2} - 6a{b^2}\]
Finally, we can simplify the solution as:
 \[{(2a + b)^3} - {(2a - b)^3} = 2{b^3} + 24{a^2}b\]
Hence the final answer is \[2{b^3} + 24{a^2}b\] .
So, the correct answer is “ \[2{b^3} + 24{a^2}b\] ”.

Note: We can also apply the following formula directly to solve the question:
 \[{(a + b)^3} = {a^3} + 3{a^2}b + 3a{b^2} + {b^3}\] and \[{(a - b)^3} = {a^3} - 3{a^2}b + 3a{b^2} - {b^3}\]
Now, \[{a^3} - {b^3}\] Should not be confused with \[{(a - b)^3}\] . \[{a^3} - {b^3}\] Can be expanded as follows:
 \[{a^3} - {b^3} = {(a - b)^3} + 3ab(a - b)\]
 \[{a^3} - {b^3} = (a - b)({a^2} + ab + {b^2})\]
Careful attention to the question is required to identify which formula is to be used.