To simplify the expression [tex]\(\sqrt{27 x^2 y^2 z}\)[/tex], we will go through the following steps:
1. Factor the expression inside the square root:
[tex]\[
27 x^2 y^2 z = 3^3 x^2 y^2 z
\][/tex]
2. Break down the square root into separate square roots for each factor:
[tex]\[
\sqrt{27 x^2 y^2 z} = \sqrt{3^3 x^2 y^2 z}
\][/tex]
3. Separate the square root into two parts:
[tex]\[
\sqrt{3^3 x^2 y^2 z} = \sqrt{3^2 \cdot 3 \cdot x^2 y^2 z}
\][/tex]
4. Simplify the square roots of the perfect squares:
[tex]\[
\sqrt{3^2 \cdot 3 \cdot x^2 y^2 z} = \sqrt{3^2} \cdot \sqrt{3 x^2 y^2 z} = 3 \cdot \sqrt{3 x^2 y^2 z}
\][/tex]
5. Recognize that the expression inside the remaining square root can be further simplified:
[tex]\[
\sqrt{3 x^2 y^2 z} = \sqrt{3} \cdot \sqrt{x^2 y^2 z}
\][/tex]
6. Simplify the square roots of the squares [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex]:
[tex]\[
\sqrt{x^2} = |x| \quad \text{and} \quad \sqrt{y^2} = |y|
\][/tex]
7. Combine the results from the square roots:
[tex]\[
\sqrt{x^2 y^2 z} = |x| \cdot |y| \cdot \sqrt{z}
\][/tex]
8. Putting it all together:
[tex]\[
3 \cdot \sqrt{3} \cdot |x| \cdot |y| \cdot \sqrt{z} = 3 \cdot |x| \cdot |y| \cdot \sqrt{3z}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\sqrt{27 x^2 y^2 z}\)[/tex] is:
[tex]\[
3 \sqrt{3 x^2 y^2 z} = 3 |x y| \sqrt{3 z}
\][/tex]
Since we know that the absolute value signs are not specified, the simplified expression to match the answer choices given is:
[tex]\[
3 \sqrt{3 x^2 y^2 z}
\][/tex]
Upon comparing this with the provided options:
1. [tex]\(3 |x y| \sqrt{3 z}\)[/tex]
2. [tex]\(9 |x y| \sqrt{3 z}\)[/tex]
3. [tex]\(3 x y \sqrt{3 z}\)[/tex]
4. [tex]\(3 \sqrt{3 x^2 y^2 z}\)[/tex]
The correct simplified form is:
[tex]\[
3 \sqrt{3 x^2 y^2 z}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{3 \sqrt{3 x^2 y^2 z}}
\][/tex]
