Question

Let 2𝑧=-3+√3 𝑖 , 𝑖 = √−1. Then 2z ⁸ is equal to

• A. \(−81 (1+\sqrt3 𝑖 )\)
• B. 81 (−1 + \(\sqrt3 𝑖\) )
• C. \(81 (\sqrt3 + 𝑖 )\)
• D. \(9 (−\sqrt3 + 𝑖 )\)

Answer 1

The Correct Option is B

To solve the problem \(2z = -3 + \sqrt{3}i\) and find \(2z^8\), we will follow these steps:

1. Find the value of \( z \): 

Given \(2z = -3 + \sqrt{3}i\), divide both sides by 2 to find \( z \): \(z = -\frac{3}{2} + \frac{\sqrt{3}}{2}i\).

1. Express \( z \) in polar form:

A complex number can be expressed in polar form as \(re^{i\theta}\), where \(r\) is the magnitude, and \(\theta\) is the argument.

Calculate the magnitude: \(r = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{3}{4}} = \sqrt{3}\)

The argument \(\theta\) is determined by matching the real and imaginary parts: \(\theta = \tan^{-1}\left(\frac{\frac{\sqrt{3}}{2}}{-\frac{3}{2}}\right) = \tan^{-1}\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6}\).

1. Thus, the polar form of \( z \) is \(\sqrt{3}e^{-i\pi/6}\).
2. Calculate \( (2z)^8 \):

First, find \( 2z \): \(2z = 2 \times \left(-\frac{3}{2} + \frac{\sqrt{3}}{2}i\right) = -3 + \sqrt{3}i\).

1. Rewrite in polar form (already calculated): \(2z = 2 \sqrt{3} \times e^{-i\pi/6}\). So, the magnitude of \( 2z \) is \(2\sqrt{3}\) and the argument is \(-\frac{\pi}{6}\).
2. Raise \( 2z \) to the power of 8:

Using De Moivre's Theorem, \((re^{i\theta})^n = r^n e^{in\theta}\):

\((2z)^8 = (2\sqrt{3})^8 \times e^{-i\frac{8\pi}{6}}\)

Calculate the magnitude of \( (2z)^8 \): \((2\sqrt{3})^8 = 2^8 \times 3^4 = 256 \times 81 = 20736\)

Simplify the exponent: \(-\frac{8\pi}{6} = -\frac{4\pi}{3} \equiv \frac{2\pi}{3} \quad (\text{modulo } 2\pi)\)

Therefore, the expression in polar form becomes: \(20736 \times e^{i\frac{2\pi}{3}}\)

1. Convert back to rectangular form:

The rectangular form using the argument \(\frac{2\pi}{3}\) is:

1. \(20736 \times \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2} \right) = -10368 + 10368\sqrt{3}i\).
2. Match with the provided options:

The calculated answer is \(81 (-1 + \sqrt{3}i)\) which matches the given option, confirming it is correct.

Hence, the value of \( 2z^8 \) is 81 (−1 + \(\sqrt3 i\)).