Number of terms free from radical sign in the expansion of ${(1 + 3^{1 / 3} + 7^{1 / 2})}^{10}$ is

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Solution

The correct option is D 8
$(1 + (3^{\frac{1}{3}} + 7^{\frac{1}{2}}))^{10}$
Let
$(3^{\frac{1}{3}} + 7^{\frac{1}{2}}) = x$
Then the above expression reduces to,
$(1 + x)^{10}$
Now writing the general term, we get
$T_{r + 1} =^{10} C_{r} x^{r}$
$=^{10} C_{r} (3^{\frac{1}{3}} + 7^{\frac{1}{2}})^{r}$ .
Hence we get one rational term for each
$r = 0, 2, 3, 4, 6, 8, 9, 10$
Hence in total 8 terms.

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730

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