# In the binomial expansion of ( cube root of 3 + square root of 2) whole root of 5 find the term which does not contain irrational expression?

general term in the expansion is ${T}_{r+1}={}^{5}C_{r}.{\left(\sqrt[3]{3}\right)}^{5-r}.{\left(\sqrt{2}\right)}^{r}$

total number of terms = 5+1=6

${T}_{r+1}={}^{5}C_{r}.{\left(3\right)}^{\frac{5-r}{3}}.{2}^{r/2}$

if the term does not contain irrational terms r/2 must be integer.

r = 0, 2, 4

and $\frac{5-r}{3}mustbeinteger$

therefore

only for r =2; (5-2)/3 = 3/3 is integer.

thus only one term does not contain irrational expression.

the term which does not contain irrational expression

$={}^{5}C_{2}.{3}^{\frac{5-2}{3}}.{2}^{2/2}\phantom{\rule{0ex}{0ex}}=\frac{5*4}{2}*3*2\phantom{\rule{0ex}{0ex}}=60$

hope this helps you

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