Here is the solution to the problems on the worksheet:

Simplify the following expressions:

1. $\sqrt{-25} = 5i$
2. $\sqrt{-324} = 18i$
3. $\sqrt{-15} = i\sqrt{15}$
4. $\sqrt{-44} = i\sqrt{44} = 2i\sqrt{11}$
5. $\sqrt{-252} = i\sqrt{252} = 6i\sqrt{7}$
6. $\sqrt{-288} = i\sqrt{288} = 12i\sqrt{2}$
7. $\sqrt{-8} \cdot \sqrt{24} = i\sqrt{8} \cdot \sqrt{24} = i\sqrt{192} = i\cdot8\sqrt{3}$
8. $\sqrt{-6} \cdot \sqrt{-12} \cdot \sqrt{-5} = i\sqrt{6} \cdot i\sqrt{12} \cdot i\sqrt{5} = -i^3\sqrt{360} = -i\cdot6\sqrt{10} = -6i\sqrt{10}$

Simplify powers of $i$:

9. $i^{28} = (i^4)^7 = 1^7 = 1$
10. $i^{49} = (i^4)^{12} \cdot i = 1^{12} \cdot i = i$

Combine imaginary numbers:

11. $15i + 4i = 19i$
12. $8i - 9i \cdot 6i = 8i - 54i^2 = 8i + 54 = 54 + 8i$

Simplify complex expressions:

13. $(-4i)^3 \cdot 2i = (-4)^3 \cdot i^3 \cdot 2i = -64 \cdot (-i) \cdot 2i = 128i^2 = -128$
14. $(2i)^5 - (i\sqrt{6})^2 = (32i^5) - (-6) = 32i \cdot i - (-6) = -32 - 6 = -38$

Would you like me to expand on any explanation for these steps or further clarify the calculations?

Here are 5 related questions to deepen your understanding:

1. What is the pattern of powers of $i$ and how can it be used to simplify expressions?
2. How do you simplify products of square roots when involving imaginary numbers?
3. How can imaginary numbers like $i$ behave similarly to real numbers in arithmetic operations?
4. How can you use factoring to simplify complex radicals like $\sqrt{-288}$?
5. What are the applications of imaginary numbers in solving quadratic equations?

Tip: Always check for patterns in imaginary numbers, such as the repetition every 4