Roots and Exponents 700 Level Manhattan

We know that the sum of the cubes of a and b is 8: a^3 + b^3 = 8. We also know that a^6 b^6 = 14. Using our knowledge of the quadratic template for the difference of two squares,
x^2 y^2 = (x + y)(x y), we can rewrite a^6 b^6 = 14 as follows:

(a^3)^2 (b^3)^2 = 14
(a^3 b^3)(a^3 + b^3) = 14

Substituting for a^3 + b^3 gives:

(a^3 b^3)(8) = 14
(a^3 b^3) = 14/8 = 7/4

Now that we have a value for a^3 + b^3 and value for a^3 b^3, we can solve for the value of a^3 using elimination.

a^3 + b^3 = 8
a^3 b^3 = 7/4

2a^3 = 39/4

Divide both sides by 2, and a^3 = 39/8.

The correct answer is C.