Answer
$-\dfrac{1}{10}$
Work Step by Step
Use the rule $a^{-m}=\left(\dfrac{1}{a}\right)^m$ to obtain:
$(-1000)^{-\frac{1}{3}}=\left(\frac{1}{-1000}\right)^{\frac{1}{3}}$
Use $-1000=(-10)^3$.
$=\left(\dfrac{1}{(-10)^3}\right)^{\frac{1}{3}}$
Use the rule $\left(\dfrac{a}{b}\right)^n=\dfrac{a^m}{b^m}$:
$=\frac{1^{\frac{1}{3}}}{\left((-10)^3\right)^{\frac{1}{3}}}$
$=\frac{1}{\left((-10)^3\right)^{\frac{1}{3}}}$
Use the rule $\left(a^m\right)^n=a^{mn}$, then simplify to obtain:
$=\dfrac{1}{(-10)^{3\cdot \frac{1}{3}}}$
$=\dfrac{1}{(-10)^{1}}$
$=-\dfrac{1}{10}$
Hence, the correct answer is $-\frac{1}{10}$.
