If a + 1/a = 2 for a nonzero real number a, what is the value of a^5 + 1/a^5?

Introduction / Context:
This is an algebraic identity problem that often appears in quantitative aptitude and algebra sections. You are given a relation involving a variable a and its reciprocal 1/a, and you must compute a higher power expression a^5 + 1/a^5. Such questions test understanding of algebraic manipulation and recognition that the given equation may uniquely determine the value of a.

Given Data / Assumptions:
- a is a nonzero real number so that 1/a is defined.
- It is given that a + 1/a = 2.
- We are required to find the value of a^5 + 1/a^5.
- All calculations are in real numbers.

Concept / Approach:
One way to handle expressions involving a and 1/a is to use algebraic identities and successive squaring. However, in this particular problem, the equation a + 1/a = 2 is very special. If we multiply both sides by a and rearrange terms, we can obtain a quadratic equation in a. Solving that quadratic shows that a has a single real value, which allows us to compute a^5 and 1/a^5 directly without heavy algebraic expansion.

Step-by-Step Solution:
Step 1: Start from the given equation a + 1/a = 2.Step 2: Multiply both sides by a to remove the denominator: a^2 + 1 = 2a.Step 3: Rearrange to standard quadratic form: a^2 - 2a + 1 = 0.Step 4: Recognize this as a perfect square: a^2 - 2a + 1 = (a - 1)^2.Step 5: Set (a - 1)^2 = 0, so a - 1 = 0 and a = 1.Step 6: Since a is nonzero and the equation has only one real root, a must be 1.Step 7: Now compute a^5 + 1/a^5 for a = 1.Step 8: a^5 = 1^5 = 1 and 1/a^5 = 1/1^5 = 1.Step 9: Therefore, a^5 + 1/a^5 = 1 + 1 = 2.

Verification / Alternative check:
Substitute a = 1 back into the original relation: a + 1/a = 1 + 1 = 2, which satisfies the given condition exactly. Since the quadratic equation derived from the original relation has only one real root, there is no other possible real value for a. Thus, any expression in a and 1/a must evaluate consistently when we use a = 1, confirming that a^5 + 1/a^5 equals 2.

Why Other Options Are Wrong:
Values such as 0, 1, 3 or 4 would arise only if a had some other value. For example, if a were 0 or -1, then a + 1/a either is not defined or does not equal 2. Since the equation a + 1/a = 2 forces a to be exactly 1, any value other than 2 for a^5 + 1/a^5 would contradict the algebraic steps performed above.

Common Pitfalls:
One mistake is to attempt long expansions using identities such as (a + 1/a)^2 and (a + 1/a)^3 without first checking whether a itself can be determined. This leads to unnecessary work. Another error is to ignore the possibility that the given relation might produce a unique solution for a. When you see simple equations like a + 1/a = 2, always consider solving directly for a before doing power expansions.

Final Answer:
The value of a^5 + 1/a^5 is 2.