To simplify the expression [tex]\(\sqrt{48 x y^2 z^3}\)[/tex], let's follow step-by-step process in detail:
### Step 1: Factorize Under the Square Root
Consider the expression inside the square root:
[tex]\[
48 x y^2 z^3
\][/tex]
### Step 2: Break Down the Constants and Variables
We can factorize [tex]\(48\)[/tex] as follows:
[tex]\[
48 = 16 \times 3
\][/tex]
So the expression becomes:
[tex]\[
\sqrt{48 x y^2 z^3} = \sqrt{16 \times 3 \times x \times y^2 \times z^3}
\][/tex]
### Step 3: Separate the Perfect Squares from the Non-perfect Squares
We know that [tex]\(16\)[/tex] is a perfect square and can be pulled out from the square root:
[tex]\[
\sqrt{16 \times 3 \times x \times y^2 \times z^3} = \sqrt{16} \times \sqrt{3 x y^2 z^3}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[
4 \sqrt{3 x y^2 z^3}
\][/tex]
### Step 4: Simplify the Remaining Expression
Now, consider the remaining expression under the square root:
[tex]\[
\sqrt{3 x y^2 z^3}
\][/tex]
We can further split this up, recognizing that both [tex]\(y^2\)[/tex] and [tex]\(z^3\)[/tex] can be simplified because [tex]\(y^2\)[/tex] is a perfect square and [tex]\(z^3\)[/tex] contains a [tex]\(z^2\)[/tex] (which is also a perfect square):
[tex]\[
\sqrt{3 x y^2 z^3} = \sqrt{3 x} \times \sqrt{y^2} \times \sqrt{z^3}
\][/tex]
Since [tex]\(\sqrt{y^2} = |y|\)[/tex] and [tex]\(\sqrt{z^3} = \sqrt{z^2 \cdot z} = |z| \sqrt{z}\)[/tex], we have:
[tex]\[
\sqrt{3 x y^2 z^3} = \sqrt{3 x} \times |y| \times |z| \times \sqrt{z}
\][/tex]
Combining all parts together:
[tex]\[
\sqrt{3 x y^2 z^3} = |y| |z| \sqrt{3 x z}
\][/tex]
### Step 5: Final Simplification
Substitute this back into our initial simplified expression:
[tex]\[
4 \sqrt{3 x y^2 z^3} = 4 \times |y| |z| \times \sqrt{3 x z}
\][/tex]
[tex]\[
4 \sqrt{3 x y^2 z^3} = 4 |y z| \sqrt{3 x z}
\][/tex]
### Conclusion
Thus, the given expression simplifies to:
[tex]\[
4 |y z| \sqrt{3 x z}
\][/tex]
Upon comparing the possible answers:
- [tex]\(4 |y z| \sqrt{3 x z}\)[/tex]
- [tex]\(4 \sqrt{3 x y^2 z^3}\)[/tex]
- [tex]\(4 y z \sqrt{3 x z}\)[/tex]
- [tex]\(16 |y z| \sqrt{3 x z}\)[/tex]
It matches exactly with the first option:
[tex]\[
\boxed{4 |y z| \sqrt{3 x z}}
\][/tex]
