The number of rational terms in the binomial expansion of ( 4^(1/4) + 5^(1/6) )¹²⁰ is _______________.

Solution

Maths sir · 4 steps

Write 4^(1/4) as √2 first, and the condition for a rational term becomes much easier.

1. Simplify the base. 4^(1/4) = (2²)^(1/4) = 2^(1/2). So the expression is (2^(1/2) + 5^(1/6))¹²⁰.

   Tᵣ₊₁ = C(120, r) 2^((120-r)/2) 5^(r/6), r = 0, 1, …, 120

2. Condition for rational. The term is rational only when both powers are whole numbers. So r/6 must be an integer, and (120-r)/2 must be an integer.
3. Combine them. If r is a multiple of 6, then r is even. Since 120 is even, 120 - r is even too, so the second condition holds automatically. We only need r = 0, 6, 12, …, 120.
4. So. The count is 120/6 + 1 = 21, so there are 21 rational terms.

The common slip: 11. Keeping 4^(1/4) as is and demanding that 120 - r be divisible by 4 forces r to be a multiple of 12, which gives 11, not 21.

The idea to keep. Reduce every base to its smallest prime power before writing the general term. Then find the values of r that make every exponent a whole number, and count them.

Answer from the NTA final answer key. Solution by PrepWiser.