The problem asks to simplify expressions involving the product of powers with the same base. We will use the property that $a^m \cdot a^n = a^{m+n}$.

1. Problem Description

The problem asks to simplify expressions involving the product of powers with the same base. We will use the property that $a^m \cdot a^n = a^{m+n}$.

2. Solution Steps

1. $7^4 \cdot 7^7$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $7^4 \cdot 7^7 = 7^{4+7} = 7^{11}$.

2. $9^6 \cdot 9^8$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $9^6 \cdot 9^8 = 9^{6+8} = 9^{14}$.

3. $2^7 \cdot 2^4$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $2^7 \cdot 2^4 = 2^{7+4} = 2^{11}$.

4. $5^6 \cdot 5^8$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $5^6 \cdot 5^8 = 5^{6+8} = 5^{14}$.

5. $7^9 \cdot 7^3$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $7^9 \cdot 7^3 = 7^{9+3} = 7^{12}$.

6. $4^7 \cdot 4^2$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $4^7 \cdot 4^2 = 4^{7+2} = 4^9$.

7. $5^8 \cdot 5^7$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $5^8 \cdot 5^7 = 5^{8+7} = 5^{15}$.

8. $4^2 \cdot 4^5$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $4^2 \cdot 4^5 = 4^{2+5} = 4^7$.

9. $9^8 \cdot 9^3$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $9^8 \cdot 9^3 = 9^{8+3} = 9^{11}$.

1

0. $5^8 \cdot 5^6$

Applying the formula $a^m \cdot a^n = a^{m+n}$, we have $5^8 \cdot 5^6 = 5^{8+6} = 5^{14}$.

3. Final Answer

1. $7^{11}$

2. $9^{14}$

3. $2^{11}$

4. $5^{14}$

5. $7^{12}$

6. $4^9$

7. $5^{15}$

8. $4^7$

9. $9^{11}$

1

0. $5^{14}$