Question

Find the value of $ (1 + i)^{10} $.

• A. \( 2^5 \)
• B. \( 32 \)
• C. \( 2^{10} \)
• D. \( 2^{5} i \)

Hint:

When solving powers of complex numbers, always try converting the complex number into polar form first and apply De Moivre's Theorem for easier computation.

Answer 1

The Correct Option is B

To solve \( (1 + i)^{10} \), we can convert the complex number \( 1 + i \) into polar form and then use De Moivre's Theorem. First, express \( 1 + i \) in polar form: \[ r = \sqrt{1^2 + 1^2} = \sqrt{2} \] \[ \theta = \tan^{-1}\left( \frac{1}{1} \right) = \frac{\pi}{4} \] So, \( 1 + i = \sqrt{2} \left( \cos \frac{\pi}{4} + i \sin \frac{\pi}{4} \right) \). 
Now, apply De Moivre's Theorem to \( (1 + i)^{10} \): \[ (1 + i)^{10} = \left( \sqrt{2} \right)^{10} \left( \cos \frac{10\pi}{4} + i \sin \frac{10\pi}{4} \right) \] Simplifying the powers and the angles: \[ \left( \sqrt{2} \right)^{10} = 2^5 = 32 \] \[ \frac{10\pi}{4} = 2\pi + \frac{\pi}{2} \] 
Thus, \( \cos \frac{10\pi}{4} = \cos \frac{\pi}{2} = 0 \), and \( \sin \frac{10\pi}{4} = \sin \frac{\pi}{2} = 1 \). 
Therefore: \[ (1 + i)^{10} = 32 \left( 0 + i \right) = 32i \] So the correct answer is (B) \( 32 \).