To evaluate $100^{-\frac{3}{2}}$, we can break it down into steps involving exponents and fractional powers. Here's the step-by-step process:

Step 1: Rewrite the Expression Using the Negative Exponent Rule

The negative exponent rule states that $a^{-b} = \frac{1}{a^b}$. Therefore:

$100^{-\frac{3}{2}} = \frac{1}{100^{\frac{3}{2}}}$

Now we need to calculate $100^{\frac{3}{2}}$.

Step 2: Rewrite $100^{\frac{3}{2}}$ as a Radical

The fractional exponent $\frac{3}{2}$ can be broken down into two steps. We can interpret $\frac{3}{2}$ as a square root (because of the denominator, 2) and then a cube (because of the numerator, 3):

$100^{\frac{3}{2}} = \left(100^{\frac{1}{2}}\right)^3$

This means we first take the square root of $100$ and then cube the result.

Step 3: Find the Square Root of 100

Since $100 = 10^2$, the square root of 100 is:

$100^{\frac{1}{2}} = 10$

Step 4: Cube the Result

Now, we take the result of the square root (which is 10) and raise it to the power of 3:

$10^3 = 1000$

So, $100^{\frac{3}{2}} = 1000$.

Step 5: Apply the Negative Exponent

Now we return to our expression from Step 1:

$100^{-\frac{3}{2}} = \frac{1}{1000}$

Final Answer

$100^{-\frac{3}{2}} = \frac{1}{1000}$

Summary

1. Rewrite $100^{-\frac{3}{2}}$ as $\frac{1}{100^{\frac{3}{2}}}$.
2. Rewrite $100^{\frac{3}{2}}$ as $\left(100^{\frac{1}{2}}\right)^3$.
3. Find the square root of 100 to get 10.
4. Cube 10 to get 1000.
5. Apply the negative exponent, resulting in $\frac{1}{1000}$.

So, $100^{-\frac{3}{2}} = \frac{1}{1000}$.

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Would you like more examples, or do you have questions on any step?

Here are five related questions you might explore:

1. What would be the value of $100^{\frac{3}{2}}$?
2. How would you evaluate $64^{-\frac{3}{2}}$?
3. How does the negative exponent rule work in general?
4. What is the value of $16^{\frac{3}{4}}$?
5. How can you evaluate $27^{-\frac{2}{3}}$?

Tip: Fractional exponents can be approached as a combination of roots and powers, making it easier to solve in steps.