To simplify the expression [tex]\(\left(-8 q^3 r^4 s^2\right)^2\)[/tex], follow these steps:
1. Identify the expression and the exponent:
[tex]\[
\left(-8 q^3 r^4 s^2\right)^2
\][/tex]
2. Apply the exponent to each term inside the parentheses:
- Coefficient: The term [tex]\(-8\)[/tex] is raised to the power of 2.
[tex]\[
(-8)^2 = 64
\][/tex]
- Variable [tex]\( q \)[/tex]: The exponent of [tex]\( q \)[/tex] is 3. When raising a power to a power, you multiply the exponents.
[tex]\[
(q^3)^2 = q^{3 \times 2} = q^6
\][/tex]
- Variable [tex]\( r \)[/tex]: The exponent of [tex]\( r \)[/tex] is 4. Similarly, raise it to the power of 2:
[tex]\[
(r^4)^2 = r^{4 \times 2} = r^8
\][/tex]
- Variable [tex]\( s \)[/tex]: The exponent of [tex]\( s \)[/tex] is 2. Raise it to the power of 2:
[tex]\[
(s^2)^2 = s^{2 \times 2} = s^4
\][/tex]
3. Combine the results:
[tex]\[
64 q^6 r^8 s^4
\][/tex]
Thus, the simplified expression is:
[tex]\[
64 q^6 r^8 s^4
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
