To determine which of the given expressions is equivalent to [tex]\(\sqrt{768 x^{19} y^{37}}\)[/tex], let's carefully analyze each of the provided options.
### Step-by-Step Analysis
First, recall that the simplification of a square root of a product can be expressed as:
[tex]\[
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\][/tex]
Applying this property to the expression [tex]\(\sqrt{768 x^{19} y^{37}}\)[/tex], we can separate the constants and variable terms under the square root:
1. Simplify the constants and variables under the square root:
[tex]\[
\sqrt{768 x^{19} y^{37}} = \sqrt{768} \cdot \sqrt{x^{19}} \cdot \sqrt{y^{37}}
\][/tex]
2. Breakdown of the constants:
[tex]\[
768 = 256 \times 3 = 16^2 \times 3 \implies \sqrt{768} = \sqrt{16^2 \times 3} = 16 \sqrt{3}
\][/tex]
3. Simplify the variable exponents:
[tex]\[
\sqrt{x^{19}} = x^{9} \sqrt{x}
\][/tex]
[tex]\[
\sqrt{y^{37}} = y^{18} \sqrt{y}
\][/tex]
Combining these simplified parts, we get:
[tex]\[
\sqrt{768 x^{19} y^{37}} = 16 \sqrt{3} \cdot x^9 \cdot \sqrt{x} \cdot y^{18} \cdot \sqrt{y} = 16 x^9 y^{18} \sqrt{3 x y}
\][/tex]
### Compare with Options
Let's check which of the given options matches this expression:
A. [tex]\(8 x^9 y^{18} \sqrt{12 x y}\)[/tex]
- This isn't matching since it has [tex]\(8\)[/tex] instead of [tex]\(16\)[/tex] and [tex]\(\sqrt{12 x y}\)[/tex] instead of [tex]\(\sqrt{3 x y}\)[/tex].
B. [tex]\(16 x^9 y^{18} \sqrt{3 x y}\)[/tex]
- This matches exactly: [tex]\(16\)[/tex], then [tex]\(x^9\)[/tex], then [tex]\(y^{18}\)[/tex], and finally [tex]\(\sqrt{3 x y}\)[/tex].
C. [tex]\(8 x^4 y^6 \sqrt{12 x^4 y}\)[/tex]
- This isn't matching since both the powers and the contents under the square root significantly differ.
D. [tex]\(16 x^4 y^6 \sqrt{3 x^4 y}\)[/tex]
- This also isn't matching as it has powers reduced ([tex]\(x^4, y^6\)[/tex]) and different contents under the square root compared to what we simplified it to.
### Conclusion
Among the provided options, the correct expression equivalent to [tex]\(\sqrt{768 x^{19} y^{37}}\)[/tex] is:
Option B: [tex]\(16 x^9 y^{18} \sqrt{3 x y}\)[/tex]
