If a + 1/a = 2, then what is the value of a^5 + 1/a^5?

Introduction / Context:
This question tests algebraic manipulation and the use of identities involving a variable and its reciprocal. When a + 1/a has a simple value, it often allows us to deduce powers like a^2, a^3 and higher without solving complicated equations repeatedly.

Given Data / Assumptions:
- We are given that a + 1/a = 2. - We assume a is non zero so that 1/a is defined. - We are asked to find a^5 + 1/a^5.

Concept / Approach:
When a + 1/a has a simple value, a common trick is to square or manipulate this relation to find a^2 + 1/a^2, a^3 + 1/a^3 and so on. However, in this particular case, the equation a + 1/a = 2 is especially simple and can be interpreted as a quadratic equation. Multiplying both sides by a gives a^2 + 1 = 2a, which can be rearranged into a standard quadratic form. Solving this will show that a is actually equal to 1, which makes higher powers easy to compute.

Step-by-Step Solution:
Step 1: Start from the given equation: a + 1/a = 2. Step 2: Multiply both sides by a (a is non zero): a^2 + 1 = 2a. Step 3: Rearrange terms to form a quadratic: a^2 - 2a + 1 = 0. Step 4: Recognize the left side as a perfect square: (a - 1)^2 = 0. Step 5: Solve for a: a - 1 = 0, so a = 1. Step 6: Now compute a^5 + 1/a^5 using a = 1. Step 7: a^5 = 1^5 = 1. Step 8: 1/a^5 = 1 / 1^5 = 1. Step 9: Therefore, a^5 + 1/a^5 = 1 + 1 = 2.

Verification / Alternative check:
We can also note that when a = 1, the original equation a + 1/a = 1 + 1 = 2 is satisfied, confirming that a = 1 is indeed a valid solution. Since the quadratic (a - 1)^2 = 0 has only one root, there is no other possible value of a. Therefore, any expression in powers of a and 1/a will be determined uniquely by a = 1. Substituting into a^5 + 1/a^5 yields 2, which is consistent.

Why Other Options Are Wrong:
Option A (0): Would require a^5 and 1/a^5 to cancel each other, which does not happen when a = 1. Option B (1): Would only occur if one of a^5 or 1/a^5 were zero or both were fractions summing to 1, which is not the case. Option C (3): Does not correspond to any logical manipulation of the given relation. Option E (4): Would require each term to be 2, which does not follow from a = 1.

Common Pitfalls:
Some students attempt to compute powers step by step using identities such as (a + 1/a)^2 or (a + 1/a)^3 without noticing that the equation reduces to a very simple quadratic. Others may forget to enforce that a is non zero. The quickest and safest way here is to solve for a directly from the quadratic form, then substitute in the required expression.

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