If x + 1/x = -2, then what is the value of x^7 + 1/x^7?

Introduction / Context:
This question involves a classical algebraic technique of using identities to find higher powers of x and 1/x, given the value of x + 1/x. Such problems are standard in algebra sections of competitive examinations.

Given Data / Assumptions:

• x + 1/x = -2.
• x is nonzero so that 1/x is defined.
• We are required to find x^7 + 1/x^7.

Concept / Approach:
When x + 1/x is known, we can often find higher symmetric expressions like x^2 + 1/x^2, x^3 + 1/x^3, and so on using identities. However, in this specific case, the value x + 1/x = -2 is special because it corresponds to a perfect square identity, which directly reveals the value of x without needing long chains of identities.

Step-by-Step Solution:
Step 1: Let S = x + 1/x. We are told S = -2.Step 2: Multiply the equation by x to remove the denominator: x^2 + 1 = -2x.Step 3: Rearrange to standard quadratic form: x^2 + 2x + 1 = 0.Step 4: Factorize this expression: x^2 + 2x + 1 = (x + 1)^2.Step 5: Set (x + 1)^2 = 0, so x + 1 = 0, giving x = -1.Step 6: Now compute x^7 + 1/x^7 for x = -1.Step 7: Since x = -1, x^7 = (-1)^7 = -1.Step 8: Also 1/x = 1/(-1) = -1, so 1/x^7 is also -1.Step 9: Therefore x^7 + 1/x^7 = -1 + (-1) = -2.

Verification / Alternative check:
We can confirm that x = -1 satisfies the original condition x + 1/x = -2: indeed, -1 + (-1) = -2. Since the quadratic equation had a repeated root, there is no other distinct value of x, so the result is unique.

Why Other Options Are Wrong:

• 1: This would require x^7 and 1/x^7 to be opposites, which does not happen for x = -1.
• -1: This would require only one term contributing, which is not the case here.
• 0: This would imply x^7 = -1/x^7, which does not hold for x = -1.

Common Pitfalls:
Some students try to compute x^2 + 1/x^2, x^3 + 1/x^3, and so on using recursive identities, which is unnecessary and time consuming when x + 1/x equals -2. Recognizing that x + 1/x = -2 immediately leads to x = -1 and greatly simplifies the problem.

Final Answer:
The value of x^7 + 1/x^7 is -2.