To simplify the expression [tex]\(\left(x y z^2\right)^4\)[/tex], we need to apply the property of exponents that states [tex]\((a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\)[/tex].
Step-by-Step Solution:
1. Identify the individual terms inside the parentheses:
The given expression is [tex]\(\left(x y z^2\right)^4\)[/tex]. Here, [tex]\(a = x\)[/tex], [tex]\(b = y\)[/tex], and [tex]\(c = z^2\)[/tex].
2. Apply the exponent to each term within the parentheses:
We raise each term inside the parentheses to the power of 4:
[tex]\[
\left(x y z^2\right)^4 = x^4 \cdot y^4 \cdot (z^2)^4
\][/tex]
3. Simplify the [tex]\(z\)[/tex] term:
For the term [tex]\((z^2)^4\)[/tex], we use the property of exponents [tex]\((z^m)^n = z^{m \cdot n}\)[/tex]:
[tex]\[
(z^2)^4 = z^{2 \cdot 4} = z^8
\][/tex]
4. Combine the results:
Now we combine all the simplified terms:
[tex]\[
x^4 \cdot y^4 \cdot z^8
\][/tex]
This means the simplified form of the expression [tex]\(\left(x y z^2\right)^4\)[/tex] is [tex]\(x^4 y^4 z^8\)[/tex].
Final Answer:
[tex]\[
\boxed{x^4 y^4 z^8}
\][/tex]
