Let's solve the given math problems step by step.

Problem 1

The expression is:

$\frac{2^{10} \cdot 2^3 \cdot 10^{-2}}{8^2 \cdot 5^2}$

First, simplify the powers of 2:

• $2^{10} \cdot 2^3 = 2^{10+3} = 2^{13}$

Next, express $8^2$ as a power of 2:

• $8 = 2^3$, so $8^2 = (2^3)^2 = 2^6$

Now, the expression becomes:

$\frac{2^{13} \cdot 10^{-2}}{2^6 \cdot 5^2}$

Simplify the powers of 2:

• $\frac{2^{13}}{2^6} = 2^{13-6} = 2^7$

Thus, the expression now is:

$2^7 \cdot \frac{10^{-2}}{5^2}$

Now express $10^{-2}$ as $\frac{1}{10^2} = \frac{1}{(2 \cdot 5)^2} = \frac{1}{2^2 \cdot 5^2}$.

Thus, the full expression is:

$2^7 \cdot \frac{1}{2^2 \cdot 5^2} \cdot \frac{1}{5^2}$

This simplifies to:

$\frac{2^{7-2}}{5^{2+2}} = \frac{2^5}{5^4}$

Therefore, the result for the first expression is:

$\frac{32}{625}$

Problem 2

The expression is:

$\frac{2^4 \cdot 5^3 \cdot 9^2}{8 \cdot 3^3 \cdot 125^{-1}}$

First, express all terms in powers of their prime factors:

• $9^2 = (3^2)^2 = 3^4$
• $8 = 2^3$
• $125^{-1} = (5^3)^{-1} = 5^{-3}$

So, the expression becomes:

$\frac{2^4 \cdot 5^3 \cdot 3^4}{2^3 \cdot 3^3 \cdot 5^{-3}}$

Combine and simplify the powers:

• For $2$: $\frac{2^4}{2^3} = 2^{4-3} = 2^1 = 2$
• For $5$: $5^{3 - (-3)} = 5^{3+3} = 5^6$
• For $3$: $\frac{3^4}{3^3} = 3^{4-3} = 3^1 = 3$

Thus, the simplified expression is:

$2 \cdot 5^6 \cdot 3$

Combine the terms:

$2 \cdot 3 \cdot 5^6 = 6 \cdot 15625 = 93750$

Therefore, the result for the second expression is $93750$.

If you have any questions or need further clarification, feel free to ask!

Here are some related questions you might find helpful:

1. How do you simplify expressions with negative exponents?
2. What are the rules for multiplying and dividing powers with the same base?
3. How can you express numbers like 8 and 125 as powers of their prime factors?
4. How do you handle expressions with different bases when simplifying?
5. What strategies can you use to check your work in simplifying exponents?
6. Why is it helpful to express all terms in prime factor form when simplifying?
7. How do you interpret a fractional exponent in an expression?
8. How can understanding exponent rules help in solving complex algebra problems?

Tip: When working with exponents, always try to express everything in terms of prime factors to make simplification easier.