Answer:
To calculate the square root of a complex number like \(3+i\), you can use the following steps:
1. **Express the Complex Number in Polar Form:**
- Find the polar form of the complex number \(3+i\). The polar form is given by:
\[ r(\cos \theta + i \sin \theta) \]
- Calculate the magnitude \(r\) and the argument \(\theta\).
The magnitude \(r\) is given by:
\[ r = \sqrt{a^2 + b^2} \]
where \(a\) and \(b\) are the real and imaginary parts of the complex number.
For \(3 + i\), \(a = 3\) and \(b = 1\), so:
\[ r = \sqrt{3^2 + 1^2} = \sqrt{10} \]
The argument \(\theta\) is given by:
\[ \theta = \arctan\left(\frac{b}{a}\right) \]
For \(3 + i\), \(\theta = \arctan\left(\frac{1}{3}\right)\).
2. **Apply the Square Root Formula:**
- Once you have the polar form, use the square root formula:
\[ \sqrt{r}(\cos \frac{\theta}{2} + i \sin \frac{\theta}{2}) \]
For \(3+i\), the square root is:
\[ \sqrt{\sqrt{10}}\left(\cos \frac{\arctan\left(\frac{1}{3}\right)}{2} + i \sin \frac{\arctan\left(\frac{1}{3}\right)}{2}\right) \]
Use a calculator to approximate the values.
This process involves converting the complex number into polar form and then applying the square root formula in polar coordinates. The result will be the square root of the given complex number.