Let's address each question individually.

Question 1

The question asks about the square root of $-20$, noting that negative numbers don’t have real square roots but can have complex square roots. The square root of $-20$ can be expressed as: $\sqrt{-20} = \sqrt{20} \cdot i = 2\sqrt{5} \cdot i$ Thus, the answer is:

D. $2\sqrt{5} i$

Question 2

Raoul needs to divide $(25 + 19i)$ by $(5 - 3i)$. To use the complex conjugate method, the first step involves multiplying both the numerator and the denominator by the complex conjugate of the denominator, $(5 + 3i)$. This gives: $\frac{25 + 19i}{5 - 3i} \cdot \frac{5 + 3i}{5 + 3i}$ Therefore, the correct answer is:

B. $\frac{(25 + 19i)(5 + 3i)}{(5 - 3i)(5 + 3i)}$

Question 3

We are asked to simplify $\frac{4 - 6i}{2i}$. Let's break down the division by rewriting the denominator:

1. Express $2i$ as $0 + 2i$.
2. Multiply both the numerator and the denominator by $-i$, which is the conjugate of $i$.

$\frac{4 - 6i}{2i} = \frac{(4 - 6i) \cdot (-i)}{2i \cdot (-i)} = \frac{-4i + 6i^2}{-2} = \frac{-4i - 6}{-2} = 3 + 2i$

So, the correct answer is:

A. $3 + 2i$

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Summary of Answers:

1. D. $2\sqrt{5} i$
2. B. $\frac{(25 + 19i)(5 + 3i)}{(5 - 3i)(5 + 3i)}$
3. A. $3 + 2i$

Would you like further clarification on any step or more detailed explanations?

Here are some additional related questions:

1. What are the properties of imaginary numbers, and why do they allow for the square roots of negative numbers?
2. How does multiplying by a complex conjugate help simplify expressions?
3. Can complex numbers be represented in polar form, and how would this change calculations?
4. What are real-life applications of complex numbers?
5. How can complex numbers be visualized on a plane?

Tip: When dividing complex numbers, always use the conjugate of the denominator to simplify the expression into a real number for the denominator.