The question asks us to evaluate $1000^{-2/3}$. Let's break it down step by step:

Step 1: Rewrite the base and exponent

We know that the general form for fractional exponents is: $a^{-b/c} = \frac{1}{(a^{b/c})}$ So: $1000^{-2/3} = \frac{1}{1000^{2/3}}$

Step 2: Simplify $1000^{2/3}$

The fractional exponent $2/3$ means we first take the cube root of $1000$, then square the result: $1000^{2/3} = \left(1000^{1/3}\right)^2$

Step 3: Cube root of 1000

The cube root of $1000$ is $10$, because $10^3 = 1000$. So: $1000^{1/3} = 10$

Step 4: Square the result

Now square $10$: $\left(1000^{1/3}\right)^2 = 10^2 = 100$

Step 5: Take the reciprocal

Finally, since the original exponent was negative: $1000^{-2/3} = \frac{1}{1000^{2/3}} = \frac{1}{100}$

Final Answer:

$1000^{-2/3} = 0.01$

Would you like further explanation or any other calculations?

Related Questions:

1. What is $27^{-2/3}$ and how does it compare to $1000^{-2/3}$?
2. What is the meaning of negative exponents in general?
3. How do fractional exponents relate to roots (e.g., cube root, square root)?
4. How would $1000^{-1/3}$ be simplified?
5. What happens if we have $1000^{2/3}$ without the negative exponent?

Tip:

To handle fractional exponents, always simplify the root (denominator of the fraction) first before applying the power (numerator of the fraction).