Simplify the expression to a, plus, b, i*a*+*bi* form:

minus, square root of, 36, end square root, plus, square root of, minus, 128, end square root, minus, square root of, 36, end square root, plus, square root of, minus, 200, end square root

To simplify the given expression to the form \( a + bi \), let's break down each part:

### Expression:
\[
-\sqrt{36} + \sqrt{-128} - \sqrt{36} + \sqrt{-200}
\]

### Step 1: Simplify each square root.

1. \(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\).
2. \(\sqrt{-128}\) can be rewritten as \(\sqrt{128} \cdot i\). 
   - \(\sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2}\), so \(\sqrt{-128} = 8\sqrt{2} \cdot i\).
3. \(\sqrt{36} = 6\), so \(-\sqrt{36} = -6\).
4. \(\sqrt{-200}\) can be rewritten as \(\sqrt{200} \cdot i\).
   - \(\sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2}\), so \(\sqrt{-200} = 10\sqrt{2} \cdot i\).

### Step 2: Substitute these simplified values back into the expression.
\[
-6 + 8\sqrt{2}i - 6 + 10\sqrt{2}i
\]

### Step 3: Combine like terms.
- Combine the real parts: \(-6 - 6 = -12\).
- Combine the imaginary parts: \(8\sqrt{2}i + 10\sqrt{2}i = 18\sqrt{2}i\).

### Final Answer:
The simplified expression in the form \( a + bi \) is:
\[
-12 + 18\sqrt{2}i
\]

Would you like any more details or have any questions?

Here are some related questions you might find interesting:
1. How do you simplify expressions involving complex numbers?
2. What are the rules for combining real and imaginary parts in complex numbers?
3. How can you find the magnitude of a complex number?
4. How can you graph a complex number on the complex plane?
5. What is the significance of the imaginary unit \(i\) in mathematics?

**Tip:** When working with square roots of negative numbers, always express them in terms of the imaginary unit \(i\).

Learn how to simplify a complex expression involving square roots and imaginary numbers to the form a + bi. This problem covers key concepts in complex numbers, including handling square roots of negative numbers and combining real and imaginary parts. Suitable for high school students studying complex numbers.

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