Question

Simplify.
Remove all perfect squares from inside the square root. Assume
\[a\] is positive.
\[\sqrt{27a}=\]

Answer 1

To simplify \(\sqrt{27a}\), we start by breaking down \(27\) into its prime factors:

\[
27 = 3^3.
\]

Thus, we can rewrite \(27a\) as follows:

\[
27a = 3^3 a.
\]

Now, we can express the square root as:

\[
\sqrt{27a} = \sqrt{3^3 \cdot a}.
\]

Next, we can separate the square root using the property \(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\):

\[
\sqrt{27a} = \sqrt{3^3} \cdot \sqrt{a}.
\]

We know that \(\sqrt{3^3} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3}\). Therefore, we can substitute this back into our expression:

\[
\sqrt{27a} = 3\sqrt{3} \cdot \sqrt{a}.
\]

This simplifies further to:

\[
\sqrt{27a} = 3\sqrt{3a}.
\]

Thus, the simplified form of \(\sqrt{27a}\) is:

\[
\boxed{3\sqrt{3a}}.
\]