fourth roots of 1-root3i
To find the fourth roots of \(1-\sqrt{3}i\), we can start by converting the number to polar form.
Let's denote the given number as \(z\):
\(z = 1 - \sqrt{3}i\)
To convert it to polar form, we need to find its magnitude (the distance from the origin) and argument (the angle it forms with the positive real axis).
Magnitude:
The magnitude of a complex number is given by the formula:
\(|z| = \sqrt{\text{Real}(z)^2 + \text{Imaginary}(z)^2}\)
For our \(z\), the magnitude is:
\(|z| = \sqrt{(1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2\)
Argument:
The argument of a complex number is given by the formula:
\(\text{arg}(z) = \text{arctan}\left(\frac{\text{Imaginary}(z)}{\text{Real}(z)}\right)\)
For our \(z\), the argument is:
\(\text{arg}(z) = \text{arctan}\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3}\)
Now, we can express \(z\) in polar form as:
\(z = 2 \cdot \text{cis}\left(-\frac{\pi}{3}\right)\)
To find the fourth roots of \(z\), we need to take the fourth root of its magnitude and divide its argument by 4.
Magnitude of the fourth root:
The magnitude of the fourth root of a complex number is given by the formula:
\( \sqrt[4]{|z|} = \sqrt[4]{2}\)
Argument of the fourth root:
The argument of the fourth root of a complex number is given by the formula:
\(\frac{\text{arg}(z)}{4}\)
So, the argument of the fourth root is:
\(\frac{-\frac{\pi}{3}}{4} = -\frac{\pi}{12}\)
Now, we can express the fourth roots of \(z\) in polar form as:
\( \sqrt[4]{2} \cdot \text{cis}\left(-\frac{\pi}{12}\right) \)
Finally, we can convert the polar form of the fourth roots back to rectangular form if needed.