The product $(a + b) (a - b) (a^2 - ab + b^2) (a^2 + ab + b^2)$ is equal to :

Question

Examoogle · Accepted Answer

The correct answer is Option D: $a^6 - b^6$. Group the conjugate-like terms: $(a+b)$ with $(a^2-ab+b^2)$ to get $(a^3+b^3)$, and $(a-b)$ with $(a^2+ab+b^2)$ to get $(a^3-b^3)$. Then apply difference of squares: $(a^3+b^3)(a^3-b^3) = (a^3)^2 - (b^3)^2$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Group the conjugate-like terms: $(a+b)$ with $(a^2-ab+b^2)$ to get $(a^3+b^3)$, and $(a-b)$ with $(a^2+ab+b^2)$ to get $(a^3-b^3)$. Then apply difference of squares: $(a^3+b^3)(a^3-b^3) = (a^3)^2 - (b^3)^2$.

Formula Used: $$(a+b)(a^2-ab+b^2) = a^3+b^3$$ $$ (a-b)(a^2+ab+b^2) = a^3-b^3 $$ $$ (x+y)(x-y) = x^2-y^2 $$

Given: We are given the algebraic expression $(a + b)(a - b)(a^2 - ab + b^2)(a^2 + ab + b^2)$.

Steps: Rearrange the expression: To simplify, regroup the terms by pairing the sum and difference of cubes factors: $[(a + b)(a^2 - ab + b^2)] 	imes [(a - b)(a^2 + ab + b^2)]$.. Apply algebraic identities: Use the standard identities for sum and difference of cubes: $(a + b)(a^2 - ab + b^2) = a^3 + b^3$ and $(a - b)(a^2 + ab + b^2) = a^3 - b^3$.. Apply difference of squares: Finally, apply the difference of squares identity $(x + y)(x - y) = x^2 - y^2$, where $x = a^3$ and $y = b^3$.

Answer: D is correct because the expression simplifies to the difference of two cubes, which further results in the difference of their sixth powers, $a^6 - b^6$.

Exam Tip: This type of simplification frequently appears in simplification-based numerical ability questions; identifying patterns in $a^2 \pm ab + b^2$ is key to avoiding long multiplications.

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Discussion (0)

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)