To solve the expression \(\sqrt{\frac{243}{64}}\), we need to take the square root of both the numerator (243) and the denominator (64) separately.

1. **Square root of the denominator:**
   - \( \sqrt{64} = 8 \), because \( 8 \times 8 = 64 \).

2. **Square root of the numerator:**
   - The square root of 243 is not a whole number, but we can simplify it by factoring:
     \[
     243 = 3 \times 81 = 3 \times 3 \times 27 = 3 \times 3 \times 3 \times 9 = 3 \times 3 \times 3 \times 3 \times 3 = 3^5
     \]
     \[
     \sqrt{243} = \sqrt{3^5} = 3^2 \times \sqrt{3} = 9\sqrt{3}
     \]

3. **Combine the results:**
   \[
   \sqrt{\frac{243}{64}} = \frac{\sqrt{243}}{\sqrt{64}} = \frac{9\sqrt{3}}{8}
   \]

So, the final simplified form of \(\sqrt{\frac{243}{64}}\) is \(\boxed{\frac{9\sqrt{3}}{8}}\).

Would you like to explore this problem further or have any related questions? Here are some related questions that might interest you:

1. How do you simplify square roots involving fractions?
2. What are the rules for simplifying square roots of larger numbers?
3. Can you explain how to rationalize a denominator with a square root?
4. How do you find the square root of a non-perfect square?
5. What are some common mistakes when dealing with square roots in algebra?

**Tip:** When simplifying square roots of fractions, always factor the numbers first to see if the square root can be simplified.

Explore how to simplify the square root of 243 over 64 with a step-by-step solution. Learn the process of simplifying square roots involving fractions and non-perfect squares. Suitable for high school students and anyone interested in advanced algebraic concepts.

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