To simplify the expression [tex]\(\left(x y z^2\right)^4\)[/tex], follow these steps:
1. Start with the given expression:
[tex]\[
\left(x y z^2\right)^4
\][/tex]
2. Apply the power of a product property, which states that [tex]\((abc)^n = a^n b^n c^n\)[/tex]. Here, [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], [tex]\(c = z^2\)[/tex], and [tex]\(n = 4\)[/tex]:
[tex]\[
\left(x y z^2\right)^4 = (x)^4 (y)^4 (z^2)^4
\][/tex]
3. Simplify each factor separately:
- [tex]\(x\)[/tex] raised to the 4th power is [tex]\(x^4\)[/tex]:
[tex]\[
(x)^4 = x^4
\][/tex]
- [tex]\(y\)[/tex] raised to the 4th power is [tex]\(y^4\)[/tex]:
[tex]\[
(y)^4 = y^4
\][/tex]
- [tex]\(z^2\)[/tex] raised to the 4th power. Use the power of a power property, which states that [tex]\((z^m)^n = z^{mn}\)[/tex]. Here [tex]\(m = 2\)[/tex] and [tex]\(n = 4\)[/tex]:
[tex]\[
(z^2)^4 = z^{2 \cdot 4} = z^8
\][/tex]
4. Combine the simplified factors:
[tex]\[
x^4 y^4 z^8
\][/tex]
Therefore, the correct simplification of [tex]\(\left(x y z^2\right)^4\)[/tex] is:
[tex]\[
x^4 y^4 z^8
\][/tex]
Hence, the correct answer is [tex]\(x^4 y^4 z^8\)[/tex].
