The problem asks us to simplify expressions involving multiplication of numbers raised to powers. We need to use the rule that when multiplying numbers with the same base, we add the exponents: $a^m \cdot a^n = a^{m+n}$.

1. Problem Description

The problem asks us to simplify expressions involving multiplication of numbers raised to powers. We need to use the rule that when multiplying numbers with the same base, we add the exponents: $a^m \cdot a^n = a^{m+n}$.

2. Solution Steps

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

We have the same base, 7, so we add the exponents:

$7^4 \cdot 7^7 = 7^{4+7} = 7^{11}$

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

We have the same base, 9, so we add the exponents:

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

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

We have the same base, 2, so we add the exponents:

$2^7 \cdot 2^4 = 2^{7+4} = 2^{11}$

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

We have the same base, 5, so we add the exponents:

$5^6 \cdot 5^8 = 5^{6+8} = 5^{14}$

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

We have the same base, 7, so we add the exponents:

$7^9 \cdot 7^3 = 7^{9+3} = 7^{12}$

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

We have the same base, 4, so we add the exponents:

$4^7 \cdot 4^2 = 4^{7+2} = 4^9$

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

We have the same base, 5, so we add the exponents:

$5^8 \cdot 5^7 = 5^{8+7} = 5^{15}$

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

We have the same base, 4, so we add the exponents:

$4^2 \cdot 4^5 = 4^{2+5} = 4^7$

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

We have the same base, 9, so we add the exponents:

$9^8 \cdot 9^3 = 9^{8+3} = 9^{11}$

1

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

We have the same base, 5, so we add the exponents:

$5^8 \cdot 5^6 = 5^{8+6} = 5^{14}$

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. $5^{14}$