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Algebra Level 5

\[ \begin{eqnarray} 1300x^3+(1729-1300\pi)x^2+(1300-1729\pi)x-1300\pi& = & 0 \\ ax^3+(b-ae)x^2+(1729-be)x-1729e & = & 0 \\ \end{eqnarray} \]

If for two complex numbers \(a, b\) there exists a common root for the two equations above, then find the value of \((1300 a b)^{\frac{1}{3}}\).

Details and Assumptions:

  • \(e= \displaystyle \lim_{y \to 0} \ (1 + y)^{ \frac {1}{y} } \approx 2.7183 \)
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