Question

Simplify the radical.

16xy6−−−−−√3
(1 point)
Responses

2y22x−−√3
2 y squared cube root of 2 x

4y2x−−√3
4 y squared cube root of x

8y22x−−√3
8 y squared cube root of 2 x

2xy22–√3

Answer 1

To simplify the expression \( \sqrt[3]{16xy^6} \), we can break it down into its factors:

1. Write \( 16 \) as \( 2^4 \).
2. We know that \( y^6 \) can be rewritten as \( (y^2)^3 \).

Thus, we can rewrite \( \sqrt[3]{16xy^6} \) in terms of its prime factors:

\[
\sqrt[3]{16xy^6} = \sqrt[3]{2^4 \cdot x \cdot (y^2)^3}
\]

Now, we can separate the cube root:

\[
\sqrt[3]{16xy^6} = \sqrt[3]{2^4} \cdot \sqrt[3]{x} \cdot \sqrt[3]{(y^2)^3}
\]

Calculating each part:

1. \( \sqrt[3]{2^4} = \sqrt[3]{16} = 2^{\frac{4}{3}} = 2^{1 + \frac{1}{3}} = 2 \cdot \sqrt[3]{2} = 2\sqrt[3]{2} \)
2. \( \sqrt[3]{x} = \sqrt[3]{x} \)
3. \( \sqrt[3]{(y^2)^3} = y^2 \)

Putting it all together, we have:

\[
\sqrt[3]{16xy^6} = 2y^2\sqrt[3]{2x}
\]

So, the simplified expression is:

\[
2y^2 \sqrt[3]{2x}
\]

Among the provided options, this can be matched with:

**2y² cube root of 2x**

The correct response would be:

**2y²√3** (Following the notation could have a misunderstanding. Clarification should be based on \( \sqrt[3]{2x} \) specifically)

The correct format should probably look like:

**2y²√(3) if matched correctly with the radical.**

This was based on assessing the options but might be censored based on context.