To simplify [tex]\(\left(-8 q^3 r^4 s^2\right)^2\)[/tex], we need to follow several steps:
1. Apply the exponent to the base and the exponents of all the variables individually:
- For the base [tex]\(-8\)[/tex], we have [tex]\((-8)^2\)[/tex].
- For [tex]\(q^3\)[/tex], we raise [tex]\(q\)[/tex] to the power [tex]\(3 \times 2\)[/tex].
- For [tex]\(r^4\)[/tex], we raise [tex]\(r\)[/tex] to the power [tex]\(4 \times 2\)[/tex].
- For [tex]\(s^2\)[/tex], we raise [tex]\(s\)[/tex] to the power [tex]\(2 \times 2\)[/tex].
2. Calculate each component:
- [tex]\((-8)^2 = 64\)[/tex] (since squaring a negative number results in a positive number).
- [tex]\(q^{3 \times 2} = q^6\)[/tex].
- [tex]\(r^{4 \times 2} = r^8\)[/tex].
- [tex]\(s^{2 \times 2} = s^4\)[/tex].
3. Combine all the results to get the simplified expression:
[tex]\[
64 q^6 r^8 s^4
\][/tex]
So, the simplified expression for [tex]\(\left(-8 q^3 r^4 s^2\right)^2\)[/tex] is:
[tex]\[
64 q^6 r^8 s^4
\][/tex]
Thus, the best answer is:
D. [tex]\(64 q^6 r^8 s^4\)[/tex]
