Express in simplest form with a rational denominator.

start fraction, 3, divided by, square root of, 54, end square root, end fraction

54​3​

We are asked to simplify the expression:

\[
\frac{3}{\sqrt{54}}
\]

### Step 1: Simplify the square root
First, break down \( \sqrt{54} \) into its prime factors:

\[
\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}
\]

So, the expression becomes:

\[
\frac{3}{3\sqrt{6}} 
\]

### Step 2: Cancel out the common factor
We can cancel the common factor of 3 in both the numerator and denominator:

\[
\frac{3}{3\sqrt{6}} = \frac{1}{\sqrt{6}}
\]

### Step 3: Rationalize the denominator
To remove the square root from the denominator, multiply both the numerator and denominator by \( \sqrt{6} \):

\[
\frac{1}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{\sqrt{6}}{6}
\]

Thus, the expression in simplest form is:

\[
\frac{\sqrt{6}}{6}
\]

Would you like further details or have any questions?

Here are 5 related questions to practice:
1. Simplify \( \frac{5}{\sqrt{8}} \).
2. Express \( \frac{4}{\sqrt{12}} \) in simplest form.
3. Rationalize the denominator of \( \frac{2}{\sqrt{5}} \).
4. Simplify \( \frac{7}{\sqrt{24}} \).
5. Express \( \frac{8}{\sqrt{50}} \) in simplest form.

**Tip:** When rationalizing a denominator, always multiply by the square root that matches the denominator!

Express in simplest form with a rational denominator: 3 divided by the square root of 54.

Learn how to simplify the expression 3 divided by the square root of 54 with a rational denominator. The solution covers breaking down square roots, simplifying expressions, and rationalizing the denominator. Suitable for students in Grades 8-10.

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