We are given the expression:

$\sqrt{16} - \sqrt{-25} - \sqrt{100} - \sqrt{-100}$

Let's simplify this step by step:

1. Simplify $\sqrt{16}$:

$\sqrt{16} = 4$

2. Simplify $\sqrt{-25}$:

Since we are taking the square root of a negative number, we will have an imaginary number. Recall that $\sqrt{-1} = i$, so: $\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$

3. Simplify $\sqrt{100}$:

$\sqrt{100} = 10$

4. Simplify $\sqrt{-100}$:

Similarly, for the square root of $-100$, we get: $\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i$

Now, putting it all together:

$4 - 5i - 10 - 10i$

Combine like terms:

• Real part: $4 - 10 = -6$
• Imaginary part: $-5i - 10i = -15i$

So, the expression simplifies to: $-6 - 15i$

This is in the form $a + bi$, where $a = -6$ and $b = -15$.

Would you like any further details or clarifications?

Here are some related questions to help you dive deeper:

1. What is the concept of imaginary numbers and why do we use $i$?
2. How can you simplify square roots of negative numbers in general?
3. What are the basic operations with complex numbers (addition, subtraction)?
4. How can you express a complex number in polar form?
5. What happens when you multiply or divide complex numbers?

Tip: When dealing with the square root of a negative number, always remember to factor out the imaginary unit $i$.