An algebra problem by Victor Loh

Algebra Level 3

Victor knows that \[2^{3} = 8 > 2^{2} = 4\] \[3^{3} = 27 > 2^{3} = 8\] \[4^{3} = 64 > 2^{4} = 16\] Since \[4^{3} - 2^{4} = 48 > 3^{3} - 2^{3} = 19 > 2^{3} - 2^{2} = 4,\] Victor concludes that for all integers \(n \geq 2\), \[n^{3} > 2^{n}.\] However, his math teacher tells him that this only holds true up till a certain value of \(n\), that is, when \(n \geq a\) where \(a\) is a positive integer, \[n^{3} \leq 2^{n}.\] Find the value of \(a\).

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