To simplify the expression \(\sqrt{1 - \sqrt{-48}}\), we need to handle the square root of a negative number, which involves complex numbers.

1. First, simplify \(\sqrt{-48}\):
   \[
   \sqrt{-48} = \sqrt{-1 \cdot 48} = \sqrt{-1} \cdot \sqrt{48} = i \cdot \sqrt{48}
   \]
   Since \(\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}\), we have:
   \[
   \sqrt{-48} = 4i\sqrt{3}
   \]

2. Substitute \(\sqrt{-48}\) back into the expression:
   \[
   \sqrt{1 - \sqrt{-48}} = \sqrt{1 - 4i\sqrt{3}}
   \]

Therefore, the simplified form of the expression is \(\sqrt{1 - 4i\sqrt{3}}\).

Would you like more details or have any questions about this process?

Here are 5 questions you can ask:
1. How do you simplify expressions involving complex numbers?
2. Can you explain the properties of imaginary numbers?
3. How do you solve quadratic equations with complex solutions?
4. What are the rules for adding and multiplying complex numbers?
5. Can you explain the geometric representation of complex numbers?

**Tip:** When dealing with square roots of negative numbers, always remember to introduce the imaginary unit \(i\), where \(i = \sqrt{-1}\).

Learn how to simplify the expression sqrt(1 - sqrt(-48)) involving complex numbers. Understand the steps to handle square roots of negative numbers and use of the imaginary unit 'i'. Suitable for advanced high school students (Grades 11-12).

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