To simplify the given expression $\left(\sqrt{81} - \sqrt{-36}\right) - \left(\sqrt{-25} + \sqrt{16}\right)$ into rectangular form $a + bi$, let's break it down step by step.

Step 1: Simplify the square roots

We will start by simplifying each square root individually:

• $\sqrt{81} = 9$
• $\sqrt{-36} = 6i$ (since $\sqrt{-1} = i$)
• $\sqrt{-25} = 5i$ (again, using $i$ for $\sqrt{-1}$)
• $\sqrt{16} = 4$

Step 2: Substitute the simplified square roots into the expression

Now substitute these results back into the expression:

$\left(9 - 6i\right) - \left(5i + 4\right)$

Step 3: Distribute the negative sign

Distribute the minus sign over the second parentheses:

$9 - 6i - 5i - 4$

Step 4: Combine like terms

• Real part: $9 - 4 = 5$
• Imaginary part: $-6i - 5i = -11i$

Step 5: Final rectangular form

Thus, the simplified expression in rectangular form is:

$5 - 11i$

Final Answer:

$\boxed{5 - 11i}$

Would you like more details, or do you have any further questions on this? Here are some related questions you might find useful:

1. How do you convert a complex number from rectangular form to polar form?
2. What is the importance of the imaginary unit $i$ in complex numbers?
3. How do you perform operations (addition, subtraction, multiplication, etc.) with complex numbers