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The question asks for the square root of two values:

1. \( \sqrt{i} \)
2. \( \sqrt{-8} \)

### Part (a): \( \sqrt{i} \)

The square root of the imaginary unit \( i \) can be determined using the following method:

1. **Express \( i \) in polar form**: \( i = e^{i\frac{\pi}{2}} \).
2. **Take the square root**: \( \sqrt{i} = \sqrt{e^{i\frac{\pi}{2}}} = e^{i\frac{\pi}{4}} \).
3. **Convert back to rectangular form**:
   \[
   \sqrt{i} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}}
   \]
   This simplifies to:
   \[
   \sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}
   \]

### Part (b): \( \sqrt{-8} \)

To find the square root of \( -8 \), we can express \( -8 \) as \( 8 \times (-1) \):

1. **Separate the negative part**: \( \sqrt{-8} = \sqrt{8} \times \sqrt{-1} \).
2. **Simplify**: \( \sqrt{8} = 2\sqrt{2} \) and \( \sqrt{-1} = i \).
3. **Combine**:
   \[
   \sqrt{-8} = 2\sqrt{2}i
   \]

### Final Answers:
- \( \sqrt{i} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)
- \( \sqrt{-8} = 2\sqrt{2}i \)

Would you like more details or have any questions?

Here are some related questions for further practice:

1. What is the square root of \( -4 \)?
2. How do you express \( i \) in exponential form?
3. What is the square root of \( 8i \)?
4. How can we represent complex numbers on an Argand diagram?
5. What is the polar form of the complex number \( -i \)?
6. How do you multiply two complex numbers in polar form?
7. What are the roots of the equation \( x^2 + 1 = 0 \)?
8. How do you add and subtract complex numbers?

**Tip:** When dealing with square roots of negative numbers, it's often useful to convert the negative number into a form that includes \( i \) to simplify calculations.

Explore the solutions to finding the square roots of i and -8 in complex numbers. This detailed guide covers the polar form of complex numbers, calculating square roots of imaginary numbers, and more. Ideal for advanced high school students and beyond.

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