The image contains several expressions with exponents to be simplified. I will solve each of them individually.

1. Problem Description

The image contains several expressions with exponents to be simplified. I will solve each of them individually.

2. Solution Steps

Problem 1: $-2^{-7} \cdot (-2)^{-5}$

Since $(-2)^{-5} = (-1)^{-5} \cdot 2^{-5} = -2^{-5}$, the expression becomes $-2^{-7} \cdot (-2^{-5})$.

$-2^{-7} \cdot (-2^{-5}) = (-1) \cdot 2^{-7} \cdot (-1) \cdot 2^{-5} = 2^{-7} \cdot 2^{-5} = 2^{-7-5} = 2^{-12} = \frac{1}{2^{12}}$.

Since $2^{10} = 1024$, we have $2^{12} = 2^{10} \cdot 2^2 = 1024 \cdot 4 = 4096$.

Therefore, the result is $\frac{1}{4096}$.

Problem 2: $7^5 \cdot 7^{-5}$

Using the rule $a^m \cdot a^n = a^{m+n}$, we have:

$7^5 \cdot 7^{-5} = 7^{5+(-5)} = 7^0 = 1$.

Problem 3: $9^6 \cdot 9^8$

Using the rule $a^m \cdot a^n = a^{m+n}$, we have:

$9^6 \cdot 9^8 = 9^{6+8} = 9^{14}$.

Problem 4: $2^3 \cdot 2^8$

Using the rule $a^m \cdot a^n = a^{m+n}$, we have:

$2^3 \cdot 2^8 = 2^{3+8} = 2^{11} = 2048$.

Problem 5: $-4^4 \cdot (-4)^{-3}$

$-4^4 \cdot (-4)^{-3} = -4^4 \cdot (-1)^{-3} \cdot 4^{-3} = -4^4 \cdot (-1) \cdot 4^{-3} = 4^4 \cdot 4^{-3} = 4^{4+(-3)} = 4^1 = 4$.

Problem 6: $5^{-2} \cdot 5^{-3}$

Using the rule $a^m \cdot a^n = a^{m+n}$, we have:

$5^{-2} \cdot 5^{-3} = 5^{-2-3} = 5^{-5} = \frac{1}{5^5}$.

$5^5 = 5^2 \cdot 5^3 = 25 \cdot 125 = 3125$.

Therefore, the result is $\frac{1}{3125}$.

Problem 7: $-3^{-4} \cdot (-3)^8$

$-3^{-4} \cdot (-3)^8 = -3^{-4} \cdot (-1)^8 \cdot 3^8 = -3^{-4} \cdot 1 \cdot 3^8 = -3^{-4} \cdot 3^8 = -3^{-4+8} = -3^4 = -81$.

Problem 8: $-4^{-8} \cdot (-4)^8$

$-4^{-8} \cdot (-4)^8 = -4^{-8} \cdot (-1)^8 \cdot 4^8 = -4^{-8} \cdot 1 \cdot 4^8 = -4^{-8} \cdot 4^8 = -4^{-8+8} = -4^0 = -1$.

Problem 9: $2^6 \cdot 2^7$

Using the rule $a^m \cdot a^n = a^{m+n}$, we have:

$2^6 \cdot 2^7 = 2^{6+7} = 2^{13} = 2^{10} \cdot 2^3 = 1024 \cdot 8 = 8192$.

Problem 10: $-2^7 \cdot (-2)^6$

$-2^7 \cdot (-2)^6 = -2^7 \cdot (-1)^6 \cdot 2^6 = -2^7 \cdot 1 \cdot 2^6 = -2^7 \cdot 2^6 = -2^{7+6} = -2^{13} = -8192$.

3. Final Answer

1. $\frac{1}{4096}$

2. $1$

3. $9^{14}$

4. $2048$

5. $4$

6. $\frac{1}{3125}$

7. $-81$

8. $-1$

9. $8192$

1

0. $-8192$