If a real number x satisfies x + 1/x = 3, use power identities to find the exact value of x^8 + 1/x^8.

Introduction / Context:
This question is a typical example of using recursive identities to compute higher powers of x and 1/x when their sum is known. Instead of trying to find x explicitly, you build up x^2 + 1/x^2, x^4 + 1/x^4 and finally x^8 + 1/x^8 using simple algebraic relations.

Given Data / Assumptions:
- x is a real number and x is non zero.
- x + 1/x = 3.
- We must find x^8 + 1/x^8.

Concept / Approach:
The method is stepwise. First compute x^2 + 1/x^2 from (x + 1/x)^2. Then compute x^4 + 1/x^4 from (x^2 + 1/x^2)^2. Finally, compute x^8 + 1/x^8 from (x^4 + 1/x^4)^2. Each step uses the identity (u + v)^2 = u^2 + 2uv + v^2 with u and v chosen appropriately.

Step-by-Step Solution:
Start with x + 1/x = 3. Square both sides: (x + 1/x)^2 = x^2 + 2 + 1/x^2 = 3^2 = 9. So x^2 + 1/x^2 = 9 - 2 = 7. Next, square x^2 + 1/x^2: (x^2 + 1/x^2)^2 = x^4 + 2 + 1/x^4. We have 7^2 = 49 = x^4 + 2 + 1/x^4. Thus x^4 + 1/x^4 = 49 - 2 = 47. Now square x^4 + 1/x^4: (x^4 + 1/x^4)^2 = x^8 + 2 + 1/x^8. We have 47^2 = 2209 = x^8 + 2 + 1/x^8. Therefore x^8 + 1/x^8 = 2209 - 2 = 2207.

Verification / Alternative check:
You could in principle solve x + 1/x = 3 as a quadratic equation x^2 - 3x + 1 = 0 and find the two possible values of x. Substituting either root into x^8 + 1/x^8 would give the same numeric result 2207. However, that route is longer than the identity based method used above.

Why Other Options Are Wrong:
Values 2201, 2203, 2211 and 2213 are close to the correct answer but differ by a few units. They are typical of arithmetic errors where the constant term 2 is added or subtracted incorrectly at one of the squaring steps. Only 2207 is consistent with the chain of identities and with any exact substitution using the quadratic roots.

Common Pitfalls:
Common mistakes include forgetting to subtract 2 when computing x^2 + 1/x^2 from (x + 1/x)^2, or using 4 instead of 2 when dealing with x^4 + 1/x^4 and x^8 + 1/x^8. Keeping track of the constant 2 at each squaring step is crucial for arriving at the correct answer.

Final Answer:
The required value of x^8 + 1/x^8 is 2207.