What is the modulus of \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?\)

Formula used:

For a complex number z = x + iy

|zⁿ| = |z|ⁿ

Where i = √-1 and

|z| = √(x² + y²)

Calculation:

Let, z = \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right) \)

This can be written as:

\(z=\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right) \)

⇒ |z²⁰⁰| = \(|\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right)^{200}| \)

Using the property of modulus:

⇒ |z|²⁰⁰ = \(|\left(\frac{{i\sqrt 3}}{2}-\frac{1}{2}\right)^{200}| \)

We know that |z| = √(x2 + y2)

⇒ |z|200 = \(\left(\sqrt{(\frac{{\sqrt3}}{2})^2+(\frac{-1}{2})^2}\right)^{200} \)

⇒ |z|200 = \((\sqrt {\frac{3}{4}+\frac{1}{4}})^{200} = 1^{200} \)

∴ \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200} =1\)