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If

\(ab = 15\) and \(a^2 + b^2 = 40\), then find the value of \((a + b)^4 - (a - b)^4\).

\(\begin{array}{|rcll|} \hline \mathbf{(a + b)^4 - (a - b)^4} &=& \left((a + b)^2\right)^2 - \left((a - b)^2\right)^2 \\ &=& \left(a^2+2ab+b^2\right)^2 - \left(a^2-2ab+b^2\right)^2 \\ &=& \left(a^2+b^2+2ab\right)^2 - \left(a^2+b^2-2ab\right)^2 \\ &=& \left(40+2*15\right)^2 - \left(40-2*15\right)^2 \\ &=& 70^2 - 10^2 \\ &=& 4900-100 \\ &=& \mathbf{4800} \\ \hline \end{array}\)