To simplify the expression [tex]\(\left(-8 q^3 r^4 s^2\right)^2\)[/tex], let's break it down step-by-step.
1. Square the coefficient:
- The coefficient in the expression is [tex]\(-8\)[/tex].
- Squaring the coefficient, we get:
[tex]\[ (-8)^2 = 64. \][/tex]
2. Square each of the variables:
- For [tex]\(q^3\)[/tex]:
[tex]\[ (q^3)^2 \Rightarrow q^{3 \cdot 2} = q^6. \][/tex]
- For [tex]\(r^4\)[/tex]:
[tex]\[ (r^4)^2 \Rightarrow r^{4 \cdot 2} = r^8. \][/tex]
- For [tex]\(s^2\)[/tex]:
[tex]\[ (s^2)^2 \Rightarrow s^{2 \cdot 2} = s^4. \][/tex]
Given these computations:
- The coefficient becomes [tex]\(64\)[/tex].
- The exponents for [tex]\(q\)[/tex], [tex]\(r\)[/tex], and [tex]\(s\)[/tex] are [tex]\(6\)[/tex], [tex]\(8\)[/tex], and [tex]\(4\)[/tex] respectively.
So the simplified expression is:
[tex]\[ 64 q^6 r^8 s^4. \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{64 q^6 r^8 s^4} \][/tex]
which corresponds to option [tex]\(D\)[/tex].
