Question

Use the identity a^3+b^3=(a+b)^3−3ab(a+b) to determine the sum of the cubes of two numbers if the sum of the two numbers is 4 and the product of the two numbers is 1.

Answer 1

Your identity says ...

... sum of cubes = (sum)³ -3(product)(sum)

... = 4³ -3·1·4

... = 64 -12 = 52

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The two numbers are 2±√3, and the sum of their cubes is indeed 52.