Let's break down the expressions and simplify them using exponent rules to compare their values:

Expressions:

1. $(7^{-2})^{-5}$
2. $\frac{7^{-20}}{7^{-10}}$
3. $7^7 \cdot 7^2$
4. $7^{-2} \cdot 7^9$

Simplification:

1. First Expression: $(7^{-2})^{-5} = 7^{-2 \times -5} = 7^{10}$

2. Second Expression: $\frac{7^{-20}}{7^{-10}} = 7^{-20 - (-10)} = 7^{-20 + 10} = 7^{-10}$

3. Third Expression: $7^7 \cdot 7^2 = 7^{7 + 2} = 7^9$

4. Fourth Expression: $7^{-2} \cdot 7^9 = 7^{-2 + 9} = 7^7$

Ordering from Least to Greatest:

Now we compare the simplified expressions in terms of their exponents:

• $7^{-10}$
• $7^7$
• $7^9$
• $7^{10}$

Therefore, the order from least to greatest is:

$\boxed{ \frac{7^{-20}}{7^{-10}} < 7^{-2} \cdot 7^9 < 7^7 \cdot 7^2 < (7^{-2})^{-5} }$

Would you like more details or have any questions?

Here are some related questions you might find useful:

1. How do you apply exponent rules to simplify complex expressions?
2. What happens when you multiply terms with the same base but different exponents?
3. What is the significance of a negative exponent?
4. How do you interpret an expression involving both a power raised to another power?
5. How do division and multiplication impact the exponents of like bases?

Tip: Always simplify expressions with the same base by adding or subtracting exponents before multiplying or dividing them.