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(x−y)(x+y)(x2+y2)(x4+y4) is equal to:
x16−y16
x8−y8
x16+y16
x8+y8
x8−y8
Use the difference of squares identity iteratively: (xn−yn)(xn+yn)=x2n−y2n.
An algebraic expression: (x - y)(x + y)(x² + y²)(x⁴ + y⁴)
(x−y)(x+y)=x2−y2
Use the difference of squares identity iteratively: (xn−yn)(xn+yn)=x2n−y2n.
Students often mistakenly assume (xn+yn)(xn+yn) or fail to recognize the pattern, sometimes leading to signs errors like x8+y8.
Step 1: Simplify the first two terms
Apply the identity (x−y)(x+y)=x2−y2.
(x−y)(x+y)=x2−y2
Step 2: Multiply by the third term
Now take the product (x2−y2)(x2+y2) and use the identity again to get x4−y4.
(x2−y2)(x2+y2)=x4−y4
Step 3: Multiply by the final term
Finally, take the product (x4−y4)(x4+y4) and use the identity one last time to arrive at the result.
(x4−y4)(x4+y4)=x8−y8
B is correct because the continuous application of the (a−b)(a+b)=a2−b2 identity results in x8−y8.
This concept of cascading differences of squares is frequently seen in competitive exams when simplifying complex radical or trigonometric expressions.
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