HMMT 2007 A4

Compute

$\frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \frac{4^{3}-1}{4^{3}+1} \cdot \frac{5^{3}-1}{5^{3}+1} \cdot \frac{6^{3}-1}{6^{3}+1}$

Dataset text NuminaMath olympiads. Quoted with source.

Show the answer

\frac{43}{63}

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