To simplify the expression [tex]\(\left(2 x^2 y^4\right)^5\left(z^2\right)^4\)[/tex] without any negative exponents, we will proceed step-by-step.
1. Apply the power rule to each part of [tex]\(\left(2 x^2 y^4\right)^5\)[/tex]:
The power rule states [tex]\((a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n\)[/tex].
So,
[tex]\[
\left(2 x^2 y^4\right)^5 = 2^5 \cdot (x^2)^5 \cdot (y^4)^5
\][/tex]
2. Simplify each term individually in [tex]\(\left(2 x^2 y^4\right)^5\)[/tex]:
[tex]\[
2^5 = 32
\][/tex]
[tex]\[
(x^2)^5 = x^{2 \cdot 5} = x^{10}
\][/tex]
[tex]\[
(y^4)^5 = y^{4 \cdot 5} = y^{20}
\][/tex]
So,
[tex]\[
\left(2 x^2 y^4\right)^5 = 32 x^{10} y^{20}
\][/tex]
3. Apply the power rule to [tex]\((z^2)^4\)[/tex]:
[tex]\[
(z^2)^4 = z^{2 \cdot 4} = z^8
\][/tex]
4. Multiply the results together:
[tex]\[
32 x^{10} y^{20} \cdot z^8 = 32 x^{10} y^{20} z^8
\][/tex]
Therefore, the simplified expression is:
[tex]\[
32 x^{10} y^{20} z^8
\][/tex]
So among the given options, the correct one is:
[tex]\[
32 x^{10} y^{20} z^8
\][/tex]
