If, a + b = 24 and \(a^2 + b^2 = 320\), then find the value of \(a^3 + b^3\).
4608
Given: \(a + b = 24\) and \(a^2 + b^2 = 320\).
First find \(ab\) using the identity \((a+b)^2 = a^2 + b^2 + 2ab\).
\(24^2 = 320 + 2ab\), so \(576 = 320 + 2ab\).
\(2ab = 256 \Rightarrow ab = 128\).
Use the identity \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\).
\(a^3 + b^3 = 24^3 - 3(128)(24)\).
\(= 13824 - 9216 = 4608\).
The value of \(a^3 + b^3\) is 4608.
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