Question

Which of the following radical expressions simplifies to the answer below?

2x15y−−−√
(1 point)
Responses

56xy2−−−−√
5 square root of 6 x y squared end root

30x2y−−−−−√
square root of 30 x squared y end root

60x2y−−−−−√
square root of 60 x squared y end root

350x2y−−−−−√

Answer 1

To simplify the expression \( \sqrt{2x^{15}y} \), we will break it down step-by-step:

1. **Factor the expression inside the square root**:
\[
\sqrt{2x^{15}y} = \sqrt{2} \cdot \sqrt{x^{15}} \cdot \sqrt{y}
\]

2. **Simplify \( \sqrt{x^{15}} \)**:
The square root of \( x^{15} \) can be simplified. Since \( x^{15} = x^{14} \cdot x^1 = (x^7)^2 \cdot x^1 \):
\[
\sqrt{x^{15}} = x^7\sqrt{x}
\]

3. **Combine all parts**:
This gives us:
\[
\sqrt{2x^{15}y} = \sqrt{2} \cdot x^7 \cdot \sqrt{y} = x^7\sqrt{2y}
\]

Now, let's examine the provided answer choices to see which one simplifies to \( x^7\sqrt{2y} \):

1. **\( \sqrt{56xy^2} \)**:
\[
\sqrt{56xy^2} = \sqrt{56} \cdot \sqrt{x} \cdot \sqrt{y^2} = \sqrt{56} \cdot \sqrt{x} \cdot y
\]
(Not \( x^7\sqrt{2y} \))

2. **\( \sqrt{30x^2y} \)**:
\[
\sqrt{30x^2y} = \sqrt{30} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{30} \cdot x \cdot \sqrt{y}
\]
(Not \( x^7\sqrt{2y} \))

3. **\( \sqrt{60x^2y} \)**:
\[
\sqrt{60x^2y} = \sqrt{60} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{60} \cdot x \cdot \sqrt{y}
\]
(Not \( x^7\sqrt{2y} \))

4. **\( \sqrt{350x^2y} \)**:
\[
\sqrt{350x^2y} = \sqrt{350} \cdot \sqrt{x^2} \cdot \sqrt{y} = \sqrt{350} \cdot x \cdot \sqrt{y}
\]
(Not \( x^7\sqrt{2y} \))

None of the options directly match \( x^7\sqrt{2y} \), but it seems there may have been oversight in transcribing or calculating the initial radical expressions.

If we want a check for radical versions, consider breaking down \( \sqrt{2x^{15}y} \) into what we went through earlier directly corresponds to those forms. Through this lens, we produced that base form and found that:

- Checking inputs might yield specific umatched values.

If asked, reevaluate the options for any direct equivalences or express as codified \( y\) values; however, among the selections without representational shifts made, please confirm if corrections or clarities needed for clarification if pursued for modification or checks in root exit formats.

In summary, simplifying \( \sqrt{2x^{15}y} = x^7\sqrt{2y} \) and doesn't match other selections provided; true answer as reformulated isn't contained within the available selections.