Question

Simplify.
Remove all perfect squares from inside the square root.
\[{\sqrt[]{180}}=\]

Answer 1

To simplify \(\sqrt{180}\), we start by factoring 180 into its prime factors.

First, we can break down 180 as follows:

\[
180 = 18 \times 10
\]
Next, we can factor 18 and 10:

\[
18 = 9 \times 2 = 3^2 \times 2
\]
\[
10 = 5 \times 2
\]

Putting it all together, we have:

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

Now, we rewrite \(180\) using its prime factors:

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

Next, we apply the property of square roots:

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

We can take the square roots of the perfect squares:

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

This simplifies to:

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

Thus, the simplified form of \(\sqrt{180}\) is:

\[
\boxed{6\sqrt{5}}
\]