Simplify the root below and leave your answer in the form  \sqrt[]{ \frac{a}{b} } 
 \sqrt[]{ \frac{405}{324} } 

We are asked to simplify \( \sqrt{\frac{405}{324}} \) and express it in the form \( \sqrt{\frac{a}{b}} \).

### Step 1: Simplify the fraction inside the square root
To simplify the square root of a fraction, first simplify the fraction \( \frac{405}{324} \).

Find the greatest common divisor (GCD) of 405 and 324. The GCD of 405 and 324 is 81.

\[
\frac{405}{324} = \frac{405 \div 81}{324 \div 81} = \frac{5}{4}
\]

### Step 2: Apply the square root to the simplified fraction
Now we take the square root of the simplified fraction:

\[
\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{\sqrt{4}} = \frac{\sqrt{5}}{2}
\]

Thus, the simplified form of \( \sqrt{\frac{405}{324}} \) is \( \frac{\sqrt{5}}{2} \).

### Final Answer
The simplified form is:

\[
\sqrt{\frac{5}{4}} = \frac{\sqrt{5}}{2}
\]

Let me know if you need any clarification!

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**Relative Questions**:
1. What happens if the GCD of the numbers is 1?
2. How can you verify the GCD using prime factorization?
3. What if the square root was an irrational number instead of a fraction?
4. How would this change if the fraction was not reducible?
5. How do you handle square roots of negative fractions?

**Tip**: Always simplify the fraction before applying any operations like square roots to make calculations easier.

Simplify the root below and leave your answer in the form \sqrt[]{ \frac{a}{b} } \sqrt[]{ \frac{405}{324} }

Learn how to simplify the square root of a fraction, \sqrt{\frac{405}{324}}, step-by-step by using properties of square roots and simplifying fractions with the greatest common divisor (GCD). This method helps students in Grades 8-10 understand simplifying radicals.

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