If x = √2 + 1, then find the exact simplified value of x^4 − 1 / x^4 expressed in terms of √2.

Introduction / Context:
This question combines algebraic manipulation with surds. It checks whether you can work with expressions involving √2, powers, and reciprocals, and then simplify to a neat exact form. Such techniques are very useful in algebra, number theory, and competitive exam problems that demand exact radical answers instead of decimals.

Given Data / Assumptions:

• x = √2 + 1.
• We need to compute x^4 − 1 / x^4.
• All operations are exact, without decimal approximation.

Concept / Approach:
Main ideas:

• Simplify step by step using powers: first find x^2, then x^4.
• Use the idea that 1 / x can be simplified by rationalising the denominator.
• After finding 1 / x, compute 1 / x^2 and 1 / x^4 similarly.
• Subtract the two exact expressions to obtain a multiple of √2.

Step-by-Step Solution:
Start with x = √2 + 1. Compute x^2 = (√2 + 1)^2 = 2 + 2√2 + 1 = 3 + 2√2. Now compute x^4 = (x^2)^2 = (3 + 2√2)^2. (3 + 2√2)^2 = 9 + 12√2 + 8 = 17 + 12√2. Find 1 / x by rationalising: 1 / (√2 + 1) = (√2 − 1) / [(√2 + 1)(√2 − 1)] = √2 − 1. Then 1 / x^2 = (√2 − 1)^2 = 2 − 2√2 + 1 = 3 − 2√2. Next 1 / x^4 = (1 / x^2)^2 = (3 − 2√2)^2 = 9 − 12√2 + 8 = 17 − 12√2. Finally compute x^4 − 1 / x^4 = (17 + 12√2) − (17 − 12√2) = 24√2.

Verification / Alternative check:
You can approximate √2 as 1.414 and calculate x ≈ 2.414. Then x^4 ≈ 2.414^4 and 1 / x^4 ≈ 1 / 2.414^4. Numerically, their difference is very close to 24 * 1.414, validating the exact symbolic result.

Why Other Options Are Wrong:
Option a: 8√2 is too small and would correspond to only one third of the correct difference.
Option b: 18√2 underestimates the true value; the algebraic computation clearly gives 24√2.
Option c and option e: 6√2 or 12√2 are both significantly smaller than the correct multiple obtained from the exact expansion.

Common Pitfalls:
Typical mistakes include squaring incorrectly, especially when handling terms like 2√2, or failing to rationalise the reciprocal properly. Another error is to attempt to approximate early with decimals, which can hide the exact pattern. Working symbolically and systematically with surds keeps the steps clear.

Final Answer:
The exact value of x^4 − 1 / x^4 is 24√2.