Question

Which expression is equivalent to 50x5y2−−−−−−√
?(1 point)
Responses

10x3y
10 x cubed y

5x2y2x−−√
5 x squared y square root of 2 x

5xy2x−−√
5 x y square root of 2 x

25x2yx−−√

Answer 1

To simplify the expression \(\sqrt{50x^5y^2}\), we can break it down into manageable parts.

1. **Factor the constant and the variables:**
\[
50 = 25 \times 2
\]
So, we rewrite:
\[
\sqrt{50x^5y^2} = \sqrt{25 \times 2 \times x^5 \times y^2}
\]

2. **Apply the square root property:**
\[
\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}
\]
Therefore:
\[
\sqrt{50x^5y^2} = \sqrt{25} \times \sqrt{2} \times \sqrt{x^5} \times \sqrt{y^2}
\]

3. **Calculate each square root:**
- \(\sqrt{25} = 5\)
- \(\sqrt{2}\) remains \(\sqrt{2}\)
- \(\sqrt{x^5} = x^{5/2} = x^2 \sqrt{x}\) (because \(x^5 = x^{4+1} = (x^2)^2 \cdot x\))
- \(\sqrt{y^2} = y\)

4. **Putting it all together:**
\[
\sqrt{50x^5y^2} = 5 \times \sqrt{2} \times x^2 \sqrt{x} \times y = 5x^2y\sqrt{2x}
\]

Now let's compare this simplified result \(5x^2y\sqrt{2x}\) with the answer choices given:

- **First option:** \(10x^3y\)
- **Second option:** \(5x^2y\sqrt{2x}\)
- **Third option:** \(5xy\sqrt{2x}\)
- **Fourth option:** \(25x^2yx\)

The expression that matches our simplification \(5x^2y\sqrt{2x}\) is the **second option**.

Therefore, the correct answer is:
\[
\boxed{5x^2y\sqrt{2x}}
\]