Question:

<p><strong>Step 1:</strong> Rewrite the expression: <span>$(\sqrt[8]{5} + \sqrt[6]{2})^{100} = (5^{1/8} + 2^{1/6})^{100}$</span></p><p><strong>Step 2:</strong> General term is <span>$T_{r+1} = \binom{100}{r}(5^{1/8})^{100-r}(2^{1/6})^r = \binom{100}{r}5^{(100-r)/8} \cdot 2^{r/6}$</span></p><p><strong>Step 3:</strong> Since 2 and 5 are coprime, <span>$T_{r+1}$</span> is rational if and only if <span>$(100-r)$</span> is a multiple of 8 AND <span>$r$</span> is a multiple of 6.</p><p><strong>Step 4:</strong> For <span>$0 \leq r \leq 100$</span>:</p><ul><li>Multiples of 6: <span>$r = 0, 6, 12, 18, \ldots, 96$</span></li><li>For <span>$(100-r)$</span> to be a multiple of 8: <span>$100-r = 0, 8, 16, 24, \ldots, 100$</span></li></ul><p><strong>Step 5:</strong> Common values: <span>$r = 12, 36, 60, 84$</span> (giving 4 rational terms)</p><p><strong>Step 6:</strong> Total terms = 101 (from <span>$r = 0$</span> to <span>$r = 100$</span>)</p><p><strong>∴ Number of irrational terms = 101 - 4 = 97</strong></p>