find square root of complex number 2i+2√3i 

To find the square root of the complex number \( 2i + 2\sqrt{3}i \), we start by expressing the complex number in standard form. Given that \( 2i + 2\sqrt{3}i \) simplifies to \( 2(1 + \sqrt{3})i \), we can proceed with the following steps.

First, recall that for any complex number \( z = a + bi \), its square roots \( \sqrt{z} \) are given by:

\[ \sqrt{z} = \pm (\sqrt{\frac{r + a}{2}} + i \cdot \text{sign}(b)\sqrt{\frac{r - a}{2}}) \]

where \( r \) is the modulus of \( z \), calculated as \( r = \sqrt{a^2 + b^2} \), and \( \text{sign}(b) \) is the sign function indicating the sign of \( b \).

For the given complex number \( 2(1 + \sqrt{3})i \):

1. Identify \( a \) and \( b \):
   - Here, \( a = 0 \)
   - \( b = 2(1 + \sqrt{3}) \)

2. Calculate the modulus \( r \):
   \[ r = \sqrt{a^2 + b^2} = \sqrt{0^2 + [2(1 + \sqrt{3})]^2} = \sqrt{4(1 + \sqrt{3})^2} = 2(1 + \sqrt{3}) \]

3. Apply the formula for the square roots:
   \[ \sqrt{z} = \pm \left( \sqrt{\frac{r + a}{2}} + i \cdot \text{sign}(b) \sqrt{\frac{r - a}{2}} \right) \]

Since \( a = 0 \) and \( r = 2(1 + \sqrt{3}) \):
   \[ \sqrt{z} = \pm \left( \sqrt{\frac{2(1 + \sqrt{3})}{2}} + i \cdot \text{sign}(2(1 + \sqrt{3})) \sqrt{\frac{2(1 + \sqrt{3})}{2}} \right) \]
   \[ \sqrt{z} = \pm \left( \sqrt{1 + \sqrt{3}} + i \sqrt{1 + \sqrt{3}} \right) \]

4. Simplify:
   \[ \sqrt{z} = \pm \left( (1 + \sqrt{3})^{1/2} + i (1 + \sqrt{3})^{1/2} \right) \]
   \[ \sqrt{z} = \pm (1 + \sqrt{3})^{1/2}(1 + i) \]

Therefore, the square roots of the complex number \( 2i + 2\sqrt{3}i \) are:
\[ \pm (1 + \sqrt{3})^{1/2}(1 + i) \]

Would you like more details or have any questions about this solution?

Here are some related questions for further exploration:

1. What is the square root of a complex number in polar form?
2. How do you find the modulus and argument of a complex number?
3. What are the properties of the square roots of a complex number?
4. How can you verify the square roots of a complex number?
5. What are the applications of complex numbers in engineering?
6. How does Euler's formula relate to complex numbers?
7. What are the cube roots of a given complex number?
8. How do you perform arithmetic operations with complex numbers?

**Tip:** Always check your result by squaring the obtained roots to ensure they match the original complex number.

Learn how to find the square root of the complex number 2i + 2√3i with a detailed step-by-step solution. This problem covers complex number theory, including the square root formula for complex numbers. Suitable for college-level students.

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