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Question
The number of irrational terms in the expansion of \[\left( 4^{1/5} + 7^{1/10} \right)^{45}\] is
Options
40
5
41
none of these
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Solution
41
\[\text{ The general term } T_{r + 1} \text{ in the given expansion is given by } \]
\[ ^{45}{}{C}_r ( 4^{1/5} )^{45 - r} ( 7^{1/10} )^r \]
\[\text{ For } T_{r + 1}\text{ to be an integer, we must have } \frac{r}{5} \text{ and } \frac{r}{10}\text { as integers i . e } . 0 \leq r \leq 45\]
\[ \therefore r = 0, 10, 20, 30 \text{ and } 40\]
\[\text{ Hence, there are 5 rational and 41, i . e . , 46 - 5, irrational terms } .\]
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