Question

In the expansion of $(\sqrt[5]{3}+\sqrt[3]{12}{)}^{15}$

• A. Number of rational terms is 3
• B. Sum of all rational terms is 58
• C. Sum of all rational terms is greater than the sum of all irrational terms
• D. Sum of all irrational terms is greater than the sum of all rational terms

Answer 1

Answer: (D)

Given binomial is $(\sqrt{5}\sqrt{3}+\sqrt[3]{2}{)}^{15}$
$\because$ The general term $T_{r+1}={}^{15}{C}_r{3}^{\frac{15-r}{5}}{2}^{\frac{r}{3}}$
$={}^{15}{C}_r{3}^{3-r/5}{2}^{r/3}$
For rational terms $r$ must be multiple of 15, so possible values of $r=0$ and
$15(\because 0\le r\le 15)$
$\therefore$ Sum of rational terms
$={}^{15}{C}_0{3}^3+{}^{15}{C}_{15}{2}^5$
$=27+32=59$
$\therefore$ The sum of all irrational terms is greater than the sum of all rational
terms.