Let's analyze the expressions and solve them step by step:

First Expression:

$9^{-2}$ Using the rule $a^{-n} = \frac{1}{a^n}$, we get: $9^{-2} = \frac{1}{9^2} = \frac{1}{81}$

Second Expression:

$9^{-1}$ Again using $a^{-n} = \frac{1}{a^n}$: $9^{-1} = \frac{1}{9}$

Thus, we compare: $\frac{1}{81} \quad \text{and} \quad \frac{1}{9}$ Clearly, $\frac{1}{81} < \frac{1}{9}$, so the first relation is: $9^{-2} < 9^{-1}$

Third Expression:

$9^{-2} \quad \text{and} \quad \left(\frac{1}{9}\right)^{-1}$ For $\left(\frac{1}{9}\right)^{-1}$, using the exponent rule for fractions, it becomes: $\left(\frac{1}{9}\right)^{-1} = 9$ We now compare: $\frac{1}{81} \quad \text{and} \quad 9$ Clearly, $\frac{1}{81} < 9$, so the second relation is: $9^{-2} < \left(\frac{1}{9}\right)^{-1}$

Fourth Expression:

$9^{-2} \quad \text{and} \quad \left(\frac{1}{9}\right)^{-2}$ For $\left(\frac{1}{9}\right)^{-2}$, it simplifies as: $\left(\frac{1}{9}\right)^{-2} = 9^2 = 81$ Comparing: $\frac{1}{81} \quad \text{and} \quad 81$ Clearly, $\frac{1}{81} < 81$, so the third relation is: $9^{-2} < \left(\frac{1}{9}\right)^{-2}$

Final Answer:

1. $9^{-2} < 9^{-1}$
2. $9^{-2} < \left(\frac{1}{9}\right)^{-1}$
3. $9^{-2} < \left(\frac{1}{9}\right)^{-2}$

Would you like a detailed explanation of any specific step or have further questions?

Here are five related questions to expand your understanding:

1. How do negative exponents affect large numbers compared to small numbers?
2. What is the general rule for handling negative exponents with fractions?
3. How do exponential expressions differ when the base is less than 1 versus when it's greater than 1?
4. What happens if we raise a fraction to a negative power and then a positive power?
5. Can the properties of exponents simplify complex algebraic expressions?

Tip: Always remember that raising a fraction to a negative exponent "flips" the fraction and raises it to a positive power.