Express in simplest radical form.
start fraction, square root of, 180, end square root, divided by, 6, end fraction


We are given the expression:

\[
\frac{\sqrt{180}}{6}
\]

### Step 1: Simplify the square root

Start by simplifying \(\sqrt{180}\). We first find the prime factorization of 180:

\[
180 = 2^2 \times 3^2 \times 5
\]

Using the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we break \(\sqrt{180}\) into:

\[
\sqrt{180} = \sqrt{2^2 \times 3^2 \times 5}
\]

Now, simplify the square roots of perfect squares:

\[
\sqrt{180} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{5} = 2 \times 3 \times \sqrt{5} = 6\sqrt{5}
\]

### Step 2: Substitute back into the expression

Now that we have simplified \(\sqrt{180} = 6\sqrt{5}\), substitute this back into the original expression:

\[
\frac{\sqrt{180}}{6} = \frac{6\sqrt{5}}{6}
\]

### Step 3: Simplify the fraction

We can now cancel the 6 in the numerator and denominator:

\[
\frac{6\sqrt{5}}{6} = \sqrt{5}
\]

### Final Answer:

\[
\sqrt{5}
\]

Would you like further explanation on any step, or do you have any additional questions? Here are a few related questions:

1. How do you simplify a square root with a coefficient in front?
2. What are some common properties of square roots that help simplify expressions?
3. How do you simplify square roots when the number inside is not a perfect square?
4. Can this method be applied to cube roots or higher roots?
5. What is the process for simplifying expressions with variables under square roots?

**Tip**: Always try to break down the number inside the square root into its prime factors first to make simplification easier!

Express in simplest radical form: start fraction, square root of, 180, end square root, divided by, 6, end fraction

Learn how to simplify the expression \frac{\sqrt{180}}{6} into its simplest radical form, with a step-by-step solution that breaks down prime factorization and fractional simplification. Suitable for students in Grades 7-9.

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