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Home » Alternatively, recall the formula for sum of divisors with \( a \geq 1 \): - 500apps

Alternatively, recall the formula for sum of divisors with \( a \geq 1 \): - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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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:

📌 a^3 + b^3 = (a+b)(a^2 - ab + b^2)

📌 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

📌 Now substitute \( a+b \), \( a^2+b^2 \), and \( ab \) into the sum of cubes formula:

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