You figured it out correctly, good job!
Nevertheless, let me answer the question, so that the homepage is clean and someone else might need some explanation.
We want to compute
[tex]\sqrt[3]{-1000p^{12}q^3}[/tex]
We can break the root of a product into the product of the roots:
[tex]\sqrt[3]{-1000p^{12}q^3} = \sqrt[3]{-1000} \cdot \sqrt[3]{p^{12}} \cdot \sqrt[3]{q^3}[/tex]
The square root of -1000 is -10, because
[tex] (-10)^3 = -1000 [/tex]
As for the exponents of the variables, taking the n-th root means to divide the exponent by n. So, in our case, we have to divide the exponents by 3:
[tex] \sqrt[3]{p^{12}} = p^{\frac{12}{3}}=p^4 \quad \sqrt[3]{q^3}=q^{\frac{3}{3}}=q [/tex]
So, the final answer is
[tex] \sqrt[3]{-1000p^{12}q^3} = -10p^4q [/tex]
