a^3 + b^3 = (a+b)(a^2 - ab + b^2)
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- \boxed{9 + 4\sqrt{2}}
- Question:** A philosopher of science is analyzing a relationship where \( a + b = 10 \) and \( a^2 + b^2 = 58 \). Determine \( a^3 + b^3 \).
- Start by using the identity for the sum of cubes:
- We know \( a + b = 10 \) and \( a^2 + b^2 = 58 \). Use the identity:
- a^2 + b^2 = (a+b)^2 - 2ab
- = 10^2 - 2ab \implies 58 = 100 - 2ab \implies 2ab = 42 \implies ab = 21
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