Question

Find the value of \[ \left| \left( \frac{1+i}{\sqrt{2}} \right)^{2024} \right|. \]

• A. 4
• B. \( 2^{1012} \)
• C. 1
• D. \( \sqrt{2} \)
• E. \( 2^{2024} \)

Hint:

When raising a complex number in polar form to a power, the modulus remains the same, and only the argument is multiplied by the exponent. In this case, the modulus of \( {cis} \, \theta \) is 1, so the modulus remains 1 regardless of the power.

Answer 1

The Correct Option is C

We are asked to find the modulus of the complex number raised to a large power.
Step 1: First, write the given complex number in polar form. 
The complex number \( 1 + i \) can be written as:
\[ 1 + i = \sqrt{1^2 + 1^2} \, {cis} \, \theta = \sqrt{2} \, {cis} \, \left( \frac{\pi}{4} \right), \] where \( {cis} \, \theta = \cos \theta + i \sin \theta \).
Step 2: Now, divide by \( \sqrt{2} \): \[ \frac{1+i}{\sqrt{2}} = \frac{\sqrt{2} \, {cis} \, \frac{\pi}{4}}{\sqrt{2}} = {cis} \, \frac{\pi}{4}. \] Step 3: We are asked to find the modulus of \( \left( \frac{1+i}{\sqrt{2}} \right)^{2024} \). 
Since the modulus of \( {cis} \, \theta \) is 1, we have: \[ \left| {cis} \, \frac{\pi}{4} \right| = 1. \] Step 4: Raising this to the power of 2024, we get: \[ \left| \left( {cis} \, \frac{\pi}{4} \right)^{2024} \right| = 1. \] 
Thus, the value of the expression is \( 1 \).
Therefore, the correct answer is option (C).