Question

Find all the values of the following :
(1) (1−√3​i)¹/³

Answer 1

The values of
`\((1 - √(3)i)^(1/3)\)` are given by the cube roots of the expression. There are three distinct cube roots, and they are:

`\[z_1 = 2^(2/3)e^(i\pi/6), \quad z_2 = 2^(2/3)e^(i\pi/2), \quad z_3 = 2^(2/3)e^(5i\pi/6)\]`

Step-by-step explanation:

To find the cube roots of
`\(1 - √(3)i\)`, we can express it in polar form and then apply De Moivre's Theorem. The polar form of
`\(1 - √(3)i\) is \(2e^(-i\pi/3)\),` where the magnitude is 2 and the argument is
`\(-\pi/3\)`.

Now, using De Moivre's Theorem, we raise the polar form to the power of
`\(1/3\):`

`\[(2e^(-i\pi/3))^(1/3) = 2^(1/3)e^(-(1/3)(i\pi/3))\]`

Simplifying the expression yields the three cube roots mentioned in the final answer.

In conclusion, the values of
`\((1 - √(3)i)^(1/3)\)` are three complex numbers with different arguments, resulting in the distinct roots
`\(z_1\), \(z_2\)`, and
`\(z_3\)`. The use of polar form and De Moivre's Theorem simplifies the calculation of these roots.