Question

Simplify the radical sqrt(28x ^ 4 * y ^ 3) ) 14x ^ 2 * 3sqrt(3); 4x ^ 2 * y * sqrt(7y); 2x ^ 2 * 3sqrt(7y)

Answer 1

To simplify the expression \(\sqrt{28x^4y^3}\), we can break down the components under the square root:

1. **Factor the radicand:**
\[
28 = 4 \cdot 7
\]
Thus, we can express the radicand as:
\[
\sqrt{28x^4y^3} = \sqrt{4 \cdot 7 \cdot x^4 \cdot y^3}
\]

2. **Simplify each factor under the square root:**
- \(\sqrt{4} = 2\)
- \(\sqrt{x^4} = x^2\)
- For \(y^3\), we can split it into \(y^2\) and \(y\):
\[
\sqrt{y^3} = \sqrt{y^2 \cdot y} = \sqrt{y^2} \cdot \sqrt{y} = y \sqrt{y}
\]

3. **Combine the simplified parts:**
\[
\sqrt{28x^4y^3} = 2 \cdot x^2 \cdot y \cdot \sqrt{7y}
\]
Therefore, we have:
\[
\sqrt{28x^4y^3} = 2x^2y\sqrt{7y}
\]

Thus, the simplified form of \(\sqrt{28x^4y^3}\) is:
\[
2x^2y \sqrt{7y}
\]

Now, let's check if this matches with the multiple choices given:

1. **\(14x^2 \cdot 3\sqrt{3} = 42x^2\sqrt{3}\)** — Not equal.
2. **\(4x^2 \cdot y \cdot \sqrt{7y}\)** — This can be simplified with coefficients; however, it does not match.
3. **\(2x^2 \cdot 3\sqrt{7y}\)** — Again, it does not match.

**Final Verification:**

Given that we simplified correctly, the answer we arrived at is \(2x^2y\sqrt{7y}\). None of the options given in the choices is a correct simplification of \(\sqrt{28x^4y^3}\).

Therefore, the complete and correct simplified form is:
\[
\sqrt{28x^4y^3} = 2x^2y\sqrt{7y}
\]