Question

The modulus and argument of \((1+i\sqrt{3})^{8}\) are respectively...

• A. \(2\) and \(2\pi /3\)
• B. \(256\) and \(7\pi /3\)
• C. \(256\) and \(2\pi /3\)
• D. \(64\) and \(4\pi /3\)

Hint:

Write 1 + i root 3 in polar form and apply De Moivre's theorem.

Answer 1

The Correct Option is C

Step 1: Modulus
The modulus of a power is the power of the modulus: $|1+i\sqrt{3}|=2$, so $|(1+i\sqrt{3})^{8}|=2^{8}=256$.

Step 2: Argument
Each factor contributes the angle $\dfrac{\pi}{3}$, so eight factors give $\dfrac{8\pi}{3}$.

Step 3: Principal value
Subtract $2\pi$: $\dfrac{8\pi}{3}-2\pi=\dfrac{2\pi}{3}$, which lies in the principal range.

Final Answer:
The modulus is 256 and the argument is 2π/3. \[ \boxed{\text{(C) }256\text{ and }2\pi /3} \]