# 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?

If the given expression is ${\left(\sqrt[3]{3}+\sqrt{2}\right)}^{5}$
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
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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