Let's break down the expression:

$\left( \sqrt{81} - \sqrt{-36} \right) - \left( \sqrt{-25} + \sqrt{16} \right)$

Step 1: Simplifying each term

1. $\sqrt{81} = 9$
2. $\sqrt{-36} = 6i$ (since $\sqrt{-1} = i$, and $\sqrt{36} = 6$)
3. $\sqrt{-25} = 5i$ (since $\sqrt{-1} = i$, and $\sqrt{25} = 5$)
4. $\sqrt{16} = 4$

Step 2: Substitute these values into the expression

$(9 - 6i) - (5i + 4)$

Step 3: Simplifying the expression

Now, distribute the negative sign:

$9 - 6i - 5i - 4$

Combine like terms:

1. Real parts: $9 - 4 = 5$
2. Imaginary parts: $-6i - 5i = -11i$

Thus, the simplified expression in rectangular form is:

$5 - 11i$

Final Answer:

The rectangular form is $5 - 11i$.

Would you like further details or clarifications?

Here are 5 related questions you might find interesting:

1. How do you simplify square roots of negative numbers?
2. What is the process for adding and subtracting complex numbers?
3. How can complex numbers be plotted on the complex plane?
4. How do you find the modulus of a complex number?
5. What is the polar form of a complex number?

Tip:

When simplifying square roots of negative numbers, always remember to factor out $i$ (the imaginary unit), where $\sqrt{-1} = i$.