The sum of all rational terms in the expansion of \(\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}\) is equal to:

Concept:

General term of (a + b)ⁿ is T^r+1 = ⁿC_r a^n-rb^r

Explanation:

the general term of \(\left(2^{\frac{1}{5}} + 5^{\frac{1}{3}}\right)^{15}\) is

\(T_{r+1} = {^{15}C_r} \left(5^{\frac{1}{3}}\right)^r \left(2^{\frac{1}{5}}\right)^{15-r}\)

       \(= {^{15}C_r} 5^{\frac{r}{3}}\cdot 2^{\frac{15-r}{5}}\)

terms will be rational if 

⇒ r = 0, 15

So, 2 rational terms are there

Sum of rational terms

= ¹⁵C₀ 2³ + ¹⁵C₁₅ (5)⁵

= 8 + 3125 = 3133

Option (1) is true.