We need to evaluate the expression:

$10^{\log_{100}(9)}$

Step 1: Express the logarithm in base 100 in terms of base 10

We know that $\log_{100}(9)$ can be rewritten using the change of base formula. The change of base formula is:

$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

where $a$ is any base. Let's use base 10. So,

$\log_{100}(9) = \frac{\log_{10}(9)}{\log_{10}(100)}$

Step 2: Simplify $\log_{10}(100)$

Since $100 = 10^2$, we know that:

$\log_{10}(100) = 2$

Step 3: Substitute and simplify

Now, we substitute this into our expression for $\log_{100}(9)$:

$\log_{100}(9) = \frac{\log_{10}(9)}{2}$

Thus, our original expression becomes:

$10^{\log_{100}(9)} = 10^{\frac{\log_{10}(9)}{2}}$

Step 4: Simplify the exponent

We can rewrite this as:

$10^{\frac{\log_{10}(9)}{2}} = \left(10^{\log_{10}(9)}\right)^{\frac{1}{2}}$

Since $10^{\log_{10}(9)} = 9$, we have:

$(10^{\log_{10}(9)})^{\frac{1}{2}} = 9^{\frac{1}{2}} = \sqrt{9} = 3$

Thus, the value of the expression is:

$\boxed{3}$

Would you like further explanation or have any questions?

Here are five related questions:

1. How do you apply the change of base formula for logarithms?
2. Why is $\log_{10}(100) = 2$?
3. How do you simplify expressions involving logarithms in different bases?
4. What is the general rule for $a^{\log_a(x)}$?
5. How can you compute logarithms for values other than powers of 10?

Tip: When working with logarithms, always check if the base is a simple power of another number to simplify calculations!