To simplify the given expression:

$\sqrt{36} - \sqrt{-4} - \sqrt{121} + \sqrt{-64}$

We'll handle each square root individually:

1. $\sqrt{36} = 6$, because 6 is the positive square root of 36.
2. $\sqrt{-4} = 2i$, because $\sqrt{4} = 2$ and the square root of a negative number introduces the imaginary unit $i$, so $\sqrt{-4} = 2i$.
3. $\sqrt{121} = 11$, because 11 is the positive square root of 121.
4. $\sqrt{-64} = 8i$, because $\sqrt{64} = 8$ and similarly, the negative sign introduces $i$, so $\sqrt{-64} = 8i$.

Now substitute these values back into the expression:

$6 - 2i - 11 + 8i$

Group the real and imaginary terms:

• Real part: $6 - 11 = -5$
• Imaginary part: $-2i + 8i = 6i$

Thus, the simplified expression in the form $a + bi$ is:

$-5 + 6i$

Would you like further details on how I arrived at this? Here are some related questions you might find helpful:

1. What is the square root of a negative number in complex form?
2. How do you add and subtract complex numbers?
3. What are the properties of the imaginary unit $i$?
4. Can you explain why we group real and imaginary parts separately in complex numbers?
5. How does the magnitude of a complex number work?

Tip: Always remember that the square root of a negative number introduces the imaginary unit $i$, where $i^2 = -1$.