Question

\[ (1 + \sqrt{2})^4 + (1 - \sqrt{2})^4 \]

• A. 34
• B. 17
• C. 51
• D. 68

Hint:

Use the binomial expansion to simplify powers of binomials and combine like terms.

Answer 1

The Correct Option is A

We can expand the terms \( (1 + \sqrt{2})^4 \) and \( (1 - \sqrt{2})^4 \) using the binomial expansion: \[ (1 + \sqrt{2})^4 = 1 + 4\sqrt{2} + 6 \times 2 + 4\sqrt{2} + 4 = 17 + 8\sqrt{2}, \] and \[ (1 - \sqrt{2})^4 = 1 - 4\sqrt{2} + 6 \times 2 - 4\sqrt{2} + 4 = 17 - 8\sqrt{2}. \] Now, adding both expressions: \[ (1 + \sqrt{2})^4 + (1 - \sqrt{2})^4 = (17 + 8\sqrt{2}) + (17 - 8\sqrt{2}) = 3(4) \]