To determine which of the given expressions is equivalent to [tex]\(\sqrt{768 x^{19} y^{37}}\)[/tex], we can start by simplifying the expression under the square root.
First, let's look at the expression:
[tex]\[
\sqrt{768 x^{19} y^{37}}
\][/tex]
We can break this down and simplify it further. The coefficient 768 can be written as:
[tex]\[
768 = 256 \times 3 = 16^2 \times 3
\][/tex]
Therefore,
[tex]\[
\sqrt{768} = \sqrt{16^2 \times 3} = 16 \sqrt{3}
\][/tex]
Now, let's look at the variables. We have [tex]\(x^{19}\)[/tex] and [tex]\(y^{37}\)[/tex]:
[tex]\[
\sqrt{x^{19} y^{37}} = x^{\frac{19}{2}} y^{\frac{37}{2}}
\][/tex]
Putting it all together:
[tex]\[
\sqrt{768 x^{19} y^{37}} = 16 \sqrt{3} \cdot x^{\frac{19}{2}} \cdot y^{\frac{37}{2}}
\][/tex]
We need to compare this simplified form with each of the given options to find a match.
Option A: [tex]\(16 x^4 y^6 \sqrt{3 x^4 y}\)[/tex]
Simplifying the square root term:
[tex]\[
\sqrt{3 x^4 y} = \sqrt{3} \cdot \sqrt{x^4} \cdot \sqrt{y} = \sqrt{3} \cdot x^2 \cdot \sqrt{y}
\][/tex]
Putting it into the expression:
[tex]\[
16 x^4 y^6 \sqrt{3 x^4 y} = 16 x^4 y^6 \cdot \sqrt{3} \cdot x^2 \cdot \sqrt{y} = 16 \sqrt{3} x^6 y^{6+\frac{1}{2}} = 16 \sqrt{3} x^6 y^{\frac{13}{2}}
\][/tex]
This does not match our simplified form, so Option A is incorrect.
Option B: [tex]\(8 x^9 y^{18} \sqrt{12 x y}\)[/tex]
Simplifying the square root term:
[tex]\[
\sqrt{12 x y} = \sqrt{12} \cdot \sqrt{x} \cdot \sqrt{y} = 2 \sqrt{3} \cdot \sqrt{x} \cdot \sqrt{y} = 2 \sqrt{3} x^{\frac{1}{2}} y^{\frac{1}{2}}
\][/tex]
Putting it into the expression:
[tex]\[
8 x^9 y^{18} \sqrt{12 x y} = 8 x^9 y^{18} \cdot 2 \sqrt{3} x^{\frac{1}{2}} y^{\frac{1}{2}} = 16 \sqrt{3} x^{9.5} y^{18.5} = 16 \sqrt{3} x^{\frac{19}{2}} y^{\frac{37}{2}}
\][/tex]
This simplified form is correct. Hence, Option B is correct.
Option C: [tex]\(8 x^4 y^6 \sqrt{12 x^4 y}\)[/tex]
Simplifying the square root term:
[tex]\[
\sqrt{12 x^4 y} = \sqrt{12} \cdot \sqrt{x^4} \cdot \sqrt{y} = 2 \sqrt{3} \cdot x^2 \cdot \sqrt{y} = 2 \sqrt{3} x^2 y^{\frac{1}{2}}
\][/tex]
Putting it into the expression:
[tex]\[
8 x^4 y^6 \sqrt{12 x^4 y} = 8 x^4 y^6 \cdot 2 \sqrt{3} x^2 y^{\frac{1}{2}} = 16 \sqrt{3} x^6 y^{6+\frac{1}{2}} = 16 \sqrt{3} x^6 y^{\frac{13}{2}}
\][/tex]
This is not the right match, so Option C is incorrect.
Option D: [tex]\(16 x^9 y^{18} \sqrt{3 x y}\)[/tex]
Simplifying the square root term:
[tex]\[
\sqrt{3 x y} = \sqrt{3} \cdot \sqrt{x} \cdot \sqrt{y} = \sqrt{3} \cdot x^{\frac{1}{2}} \cdot y^{\frac{1}{2}}
\][/tex]
Putting it into the expression:
[tex]\[
16 x^9 y^{18} \sqrt{3 x y} = 16 x^9 y^{18} \cdot \sqrt{3} x^{\frac{1}{2}} y^{\frac{1}{2}} = 16 \sqrt{3} x^{9.5} y^{18.5} = 16 \sqrt{3} x^{\frac{19}{2}} y^{\frac{37}{2}}
\][/tex]
This simplified form is correct. Hence, Option D is correct.
However, given that both Options B and D simplify correctly, and there should be only one correct option as per typical multiple-choice standards, we conclude that no option is designed to match our simplification exclusively.
Therefore, based on the comparison:
[tex]\[
\boxed{\text{None}}
\][/tex]
