To solve this problem, we need to simplify the given expression \(\sqrt{768 x^{19} y^{37}}\) and compare it with the provided options to identify the equivalent expression.
The given expression is:
[tex]\[
\sqrt{768 x^{19} y^{37}}
\][/tex]
First, let's simplify \(\sqrt{768 x^{19} y^{37}}\):
1. Break down the components inside the square root:
- Factor 768 into its prime factors: \(768 = 2^8 \times 3\).
- The given expression inside the square root is \(768 x^{19} y^{37}\).
2. Write the expression using the prime factorization:
[tex]\[
\sqrt{768 x^{19} y^{37}} = \sqrt{2^8 \times 3 \times x^{19} \times y^{37}}
\][/tex]
3. Simplify the square root:
- Use the property \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\).
- Separate the terms inside the square root:
[tex]\[
\sqrt{2^8 \times 3 \times x^{19} \times y^{37}} = \sqrt{2^8} \times \sqrt{3} \times \sqrt{x^{19}} \times \sqrt{y^{37}}
\][/tex]
- Simplify each part:
[tex]\[
\sqrt{2^8} = 2^4 = 16
\][/tex]
[tex]\[
\sqrt{3} = \sqrt{3}
\][/tex]
[tex]\[
\sqrt{x^{19}} = x^{19/2} = x^{9.5}
\][/tex]
[tex]\[
\sqrt{y^{37}} = y^{37/2} = y^{18.5}
\][/tex]
Combining these results gives:
[tex]\[
\sqrt{768 x^{19} y^{37}} = 16 \sqrt{3} \times x^{9.5} \times y^{18.5}
\][/tex]
[tex]\[
= 16 \sqrt{3} \times x^{9+\frac{1}{2}} \times y^{18+\frac{1}{2}}
\][/tex]
[tex]\[
= 16 \sqrt{3} \times x^9 x^{0.5} \times y^{18} y^{0.5}
\][/tex]
[tex]\[
= 16 \sqrt{3} \times x^9 y^{18} \sqrt{x y}
\][/tex]
Comparing the simplified form with the provided options, we see that:
- Option A is \(8 x^9 y^{18} \sqrt{12 x y} \).
- Option B is \(16 x^4 y^6 \sqrt{3 x^4 y}\).
- Option C is \(8 x^4 y^6 \sqrt{12 x^4 y}\).
- Option D is \(16 x^9 y^{18} \sqrt{3 x y}\).
The correct match is:
[tex]\[
16 x^9 y^{18} \sqrt{3 x y}
\][/tex]
Thus, the correct option is:
[tex]\[
\boxed{D}
\][/tex]
