https://file.aiquickdraw.com/mathbot/file/1727464461613efvpa2ol.png

The mathematical expression in the image is:

\[
\sqrt{1 + \frac{\sqrt{3}}{2}}
\]

### Steps to Simplify:
1. First, simplify the expression under the square root: \( 1 + \frac{\sqrt{3}}{2} \).
2. Combine \( 1 \) and \( \frac{\sqrt{3}}{2} \) into a single fraction:
   \[
   1 + \frac{\sqrt{3}}{2} = \frac{2}{2} + \frac{\sqrt{3}}{2} = \frac{2 + \sqrt{3}}{2}
   \]
3. Now substitute this back into the original square root:
   \[
   \sqrt{\frac{2 + \sqrt{3}}{2}}
   \]
4. Finally, you can simplify this expression by recognizing that the square root of a fraction is the square root of the numerator divided by the square root of the denominator:
   \[
   \frac{\sqrt{2 + \sqrt{3}}}{\sqrt{2}}
   \]

This is the simplified form of the expression.

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Would you like further details on any step?

### Related Questions:
1. What are the properties of square roots and how are they applied in simplifying expressions?
2. How do you simplify complex fractions involving square roots?
3. Can we approximate the numerical value of this expression?
4. How do rational and irrational numbers interact in a fraction?
5. What is the geometric interpretation of such expressions?

### Tip:
When simplifying expressions under square roots, always look for opportunities to combine terms by factoring or finding a common denominator.

What is the simplified form of the expression \( \sqrt{1 + \frac{\sqrt{3}}{2}} \)?

Learn how to simplify the square root expression \( \sqrt{1 + \frac{\sqrt{3}}{2}} \) with step-by-step explanations. This problem covers key concepts such as radical simplification, fractional operations, and properties of square roots, suitable for students in Grades 9-11.

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