To determine the cube root of the expression [tex]\(-1,000 p^{12} q^3\)[/tex], we need to take the cube root of each individual component separately. Let's break it down step-by-step:
### Step-by-Step Solution:
1. Cube Root of [tex]\(-1,000\)[/tex]:
[tex]\[
\sqrt[3]{-1,000} = -10
\][/tex]
This is because [tex]\((-10)^3 = -1,000\)[/tex].
2. Cube Root of [tex]\(p^{12}\)[/tex]:
[tex]\[
\sqrt[3]{p^{12}} = p^{4}
\][/tex]
This is because [tex]\((p^4)^3 = p^{12}\)[/tex].
3. Cube Root of [tex]\(q^3\)[/tex]:
[tex]\[
\sqrt[3]{q^3} = q
\][/tex]
This is because [tex]\(q^3\)[/tex] raised to the [tex]\(1/3\)[/tex] power is [tex]\(q\)[/tex].
### Combining the Results:
Now, we combine the cube roots of the individual components:
[tex]\[
\sqrt[3]{-1,000 p^{12} q^3} = \sqrt[3]{-1,000} \cdot \sqrt[3]{p^{12}} \cdot \sqrt[3]{q^3} = -10 \cdot p^4 \cdot q
\][/tex]
### Final Answer:
Thus, the cube root of [tex]\(-1,000 p^{12} q^3\)[/tex] is:
[tex]\[
-10 p^4 q
\][/tex]
So the correct answer is:
[tex]\[
\boxed{-10 p^4 q}
\][/tex]
