Let's simplify the given expression step by step.

Given expression:

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

Step 1: Evaluate the square roots

• $\sqrt{81} = 9$
• $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$
• $\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$
• $\sqrt{16} = 4$

Now, substituting these values into the expression:

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

Step 2: Distribute $bi$

$a + (9bi - 6bi^2) - 5i - 4$

Since $i^2 = -1$, we substitute:

$a + (9bi + 6b) - 5i - 4$

Rewriting in standard form:

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

Final Answer:

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

Would you like more details or a different approach?

Related Questions:

1. How do you convert between rectangular and polar form?
2. What is the significance of imaginary numbers in complex analysis?
3. How do you simplify expressions with complex numbers?
4. How do you multiply and divide complex numbers?
5. How do you plot complex numbers on the complex plane?

Tip: Always simplify square roots of negative numbers using $i$ before performing operations.