Question

Simplify.
Remove all perfect squares from inside the square roots. Assume
\[x\] and
\[y\] are positive.
\[\sqrt{8x^3y^2}=\]

Answer 1

To simplify \(\sqrt{8x^3y^2}\), we start by factoring out the perfect squares from inside the square root.

1. First, we can factor \(8\) as \(4 \times 2\), which gives us the perfect square \(4\).
2. Next, we can separate the powers of \(x\) and \(y\):
- \(x^3\) can be written as \(x^2 \cdot x\), where \(x^2\) is a perfect square.
- \(y^2\) is already a perfect square.

Now we can rewrite the expression under the square root:

\[
\sqrt{8x^3y^2} = \sqrt{4 \cdot 2 \cdot x^2 \cdot x \cdot y^2}
\]

Next, we use the property of square roots that states \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\):

\[
\sqrt{8x^3y^2} = \sqrt{4} \cdot \sqrt{2} \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{y^2}
\]

We know the square roots of the perfect squares:

- \(\sqrt{4} = 2\)
- \(\sqrt{x^2} = x\)
- \(\sqrt{y^2} = y\)

Putting it all together, we have:

\[
\sqrt{8x^3y^2} = 2 \cdot \sqrt{2} \cdot x \cdot \sqrt{x} \cdot y
\]

Rearranging it, we get:

\[
\sqrt{8x^3y^2} = 2xy\sqrt{2x}
\]

Thus, the simplified form is:

\[
\boxed{2xy\sqrt{2x}}
\]