Sure! Let's simplify the given expression [tex]\(\left(16 x^4 y z^6\right)^{\frac{1}{2}}\)[/tex] step-by-step as a radical expression.
The expression inside the parentheses is [tex]\(16 x^4 y z^6\)[/tex]. We need to take the square root of this entire expression, which is equivalent to raising it to the power of [tex]\(\frac{1}{2}\)[/tex].
First, we will take the square root of each component individually:
1. Square root of [tex]\(16\)[/tex]:
[tex]\[ \sqrt{16} = 4 \][/tex]
2. Square root of [tex]\(x^4\)[/tex]:
[tex]\[ \sqrt{x^4} = x^{\frac{4}{2}} = x^2 \][/tex]
3. Square root of [tex]\(y\)[/tex]:
[tex]\[ \sqrt{y} \][/tex]
4. Square root of [tex]\(z^6\)[/tex]:
[tex]\[ \sqrt{z^6} = z^{\frac{6}{2}} = z^3 \][/tex]
Now, we combine these results:
[tex]\[ \sqrt{16 x^4 y z^6} = \sqrt{16} \cdot \sqrt{x^4} \cdot \sqrt{y} \cdot \sqrt{z^6} \][/tex]
[tex]\[ = 4 \cdot x^2 \cdot \sqrt{y} \cdot z^3 \][/tex]
Putting it together in a simplified form:
[tex]\[ 4 x^2 z^3 \sqrt{y} \][/tex]
So the correct simplified radical expression is:
[tex]\[ \boxed{4 x^2 z^3 \sqrt{y}} \][/tex]
