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  • The sum of rational term(s) in <img alt="(\sqrt{3}+\sqrt[3]{2}+\sqrt[4]{5})^8" src="https://entrancecorner.oncodecogs.com/gif.latex?%28%5Csqrt%7B3%7D&plus;%5Csqrt%5B3%5D%7B2%7D&plus;%5

The sum of rational term(s) in (\sqrt{3}+\sqrt[3]{2}+\sqrt[4]{5})^8 is equal to

 

Option: 1

3150
 


Option: 2

336
 


Option: 3

3486
 


Option: 4

3592


Answers (1)

best_answer

The general term in the expansion of (\sqrt{3}+\sqrt[3]{2}+\sqrt[4]{5})^8 is \frac{8 !}{r ! s ! t !}
(3)^{r / 2}
(2)^{s / 3}
(5)^{t / 4}, where r+s+t=8
For rational terms, we must have, r=0,2,4,6,8, s=0,3, 6 and t=0,4,8
For t=0, r=8, s=0
For t=0, r=2, s=6
For t=4, r=4, s=0
For t=8, r=0, s=0

\begin{aligned} & \therefore \quad \text { Sum of rational terms }=\frac{8 !}{2 ! 6 ! 0 !} \times 3 \times 2^2 \times 5^0 \\ & +\frac{8 !}{4 ! 0 ! 4 !} \times 3^2 \times 2^0 \times 5+\frac{8 !}{0 ! 0 ! 8 !} \times 3^0 \times 2^0 \times 5^2 \\ & +\frac{8 !}{8 ! 0 ! 0 !} \times 3^4 \times 2^0 \times 5^0 \\ & =3592 \end{aligned}

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Rishi

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