If x^(1/4) + x^(−1/4) = 2 for a positive real number x, then what is the value of x^81 + 1/x^81?

Introduction / Context:
This problem tests exponent manipulation and the use of a simple but important inequality: for a positive real t, the expression t + 1/t has a minimum of 2 when t = 1. The equation x^(1/4) + x^(−1/4) = 2 is of this form and forces x^(1/4) to be equal to 1. Once x is found, evaluating very high powers like x^81 + 1/x^81 becomes trivial.

Given Data / Assumptions:

x^(1/4) + x^(−1/4) = 2.
x is a positive real number (so its real fourth root is defined and positive).
The required value is x^81 + 1/x^81.

Concept / Approach:
Let t = x^(1/4). Then x^(−1/4) is 1/t. The equation becomes t + 1/t = 2. For positive t, this equation has a unique solution t = 1, since t + 1/t is minimized at 2 and increases for t not equal to 1. Once t is known, x = t^4 can be computed and then used to evaluate x^81 and 1/x^81.

Step-by-Step Solution:
Let t = x^(1/4). Then x^(−1/4) = 1/t.The equation becomes t + 1/t = 2.Multiply both sides by t (t is positive and non zero): t^2 + 1 = 2t.Rearrange: t^2 − 2t + 1 = 0.This factors as (t − 1)^2 = 0.Hence t = 1.Recall that t = x^(1/4), so x^(1/4) = 1 implies x = 1^4 = 1.Now compute x^81 + 1/x^81: since x = 1, we have x^81 = 1 and 1/x^81 = 1.Therefore x^81 + 1/x^81 = 1 + 1 = 2.

Verification / Alternative check:
The function f(t) = t + 1/t for t > 0 has derivative f'(t) > 0, so it has a global minimum at t = 1, where it equals 2. This means that the equation t + 1/t = 2 has only the solution t = 1 in positive reals. Since we explicitly assume x is positive, t is positive and therefore t = 1 is the only possibility. This confirms that x must be 1 and that any other value for x^81 + 1/x^81 would contradict the original equation.

Why Other Options Are Wrong:
The values −2, 0, 1, and 4 would arise only if x were complex or negative in a way that yields multiple fourth roots, or if the algebraic reasoning were incorrect. Under the condition that x is positive, the only acceptable solution is x = 1, which forces x^81 + 1/x^81 to equal 2. Therefore all other options are ruled out.

Common Pitfalls:
One common mistake is to treat x^(1/4) as x/4 or to misinterpret the exponents. Another is to attempt to directly raise the equation to high powers without first solving for x. This leads to very large exponents and unnecessary complexity. Introducing t = x^(1/4) and using the simple quadratic equation is the clean and reliable method.

Final Answer:
The value of x^81 + 1/x^81 is 2.