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.