To simplify the expression [tex]\(\sqrt{3xy} - 2\sqrt{27x^3y} + 4\sqrt{3xy}\)[/tex], we'll analyze each term and combine like terms where possible.
1. Identify Like Terms
- The first term is [tex]\(\sqrt{3xy}\)[/tex].
- The second term is [tex]\(-2\sqrt{27x^3y}\)[/tex].
- The third term is [tex]\(4\sqrt{3xy}\)[/tex].
2. Simplify Each Term
[tex]\[
\sqrt{3xy} \text{ remains as } \sqrt{3xy}.
\][/tex]
Next, simplify [tex]\(-2\sqrt{27x^3y}\)[/tex]:
[tex]\[
-2\sqrt{27x^3y} = -2\sqrt{(3^3)x^3y} = -2\sqrt{3^3}\sqrt{x^3}\sqrt{y} = -2 \cdot 3\sqrt{3}\cdot x^{3/2}\cdot \sqrt{y} = -6x^{3/2}\sqrt{3y}
\][/tex]
Combine [tex]\(\sqrt{3xy}\)[/tex] and [tex]\(4\sqrt{3xy}\)[/tex]:
[tex]\[
\sqrt{3xy} + 4\sqrt{3xy} = 5\sqrt{3xy}
\][/tex]
3. Combine Like Terms
Combining all the simplified terms, we get:
[tex]\[
5\sqrt{3xy} - 6x^{3/2}\sqrt{3y}
\][/tex]
This can be factored as:
[tex]\[
\sqrt{3}(5\sqrt{xy} - 6x\sqrt{x y})
\][/tex]
Thus, the fully simplified form of [tex]\(\sqrt{3xy} - 2\sqrt{27x^3y} + 4\sqrt{3xy}\)[/tex] is:
[tex]\[
\sqrt{3}(5\sqrt{xy} - 6x\sqrt{xy})
\][/tex]
Comparing this with the given choices:
- [tex]\(5 \sqrt{3xy} - 2 \sqrt{27 x^3 y}\)[/tex]
- [tex]\(5 \sqrt{3xy} - 6 x \sqrt{3xy}\)[/tex]
- [tex]\(\sqrt{3xy} - 2 \sqrt{27 x^3 y} + 4 \sqrt{3xy}\)[/tex]
- [tex]\(-\sqrt{3xy}\)[/tex]
The correct simplified expression is:
[tex]\[
5 \sqrt{3xy} - 6 x \sqrt{3xy}
\][/tex]
Therefore, the correct answer is:
[tex]\[
5 \sqrt{3xy} - 6 x \sqrt{3xy}
\][/tex]
