Question
Download Solution PDFWhat is the modulus of \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?\)
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFFormula 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 = \((\sqrt {\frac{3}{4}+\frac{1}{4}})^{200} = 1^{200} \)
∴ \(\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200} =1\)
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