Answer :
Let's go through each of these exponent calculations step-by-step and get the results.
### 5. Multiplying Powers with the Same Base
When multiplying powers with the same base, we add the exponents according to the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
1. [tex]\( 3^3 \times 3^3 \)[/tex]:
[tex]\[ 3^3 \times 3^3 = 3^{3+3} = 3^6 = 729 \][/tex]
2. [tex]\( 2^2 \times 2^2 \)[/tex]:
[tex]\[ 2^2 \times 2^2 = 2^{2+2} = 2^4 = 16 \][/tex]
### 6. Dividing Powers with the Same Base
When dividing powers with the same base, we subtract the exponents according to the rule [tex]\(a^m \div a^n = a^{m-n}\)[/tex]:
1. [tex]\( 7^4 \div 7^2 \)[/tex]:
[tex]\[ 7^4 \div 7^2 = 7^{4-2} = 7^2 = 49 \, (\text{or } 49.0) \][/tex]
2. [tex]\( 9^5 \div 9^3 \)[/tex]:
[tex]\[ 9^5 \div 9^3 = 9^{5-3} = 9^2 = 81 \, (\text{or } 81.0) \][/tex]
3. [tex]\( 16^4 \div 16^2 \)[/tex]:
[tex]\[ 16^4 \div 16^2 = 16^{4-2} = 16^2 = 256 \, (\text{or } 256.0) \][/tex]
### 7. Multiplying Powers with the Same Base (again)
1. [tex]\( 6^4 \times 6^1 \)[/tex]:
[tex]\[ 6^4 \times 6^1 = 6^{4+1} = 6^5 = 7776 \][/tex]
2. [tex]\( 4^4 \times 4^2 \)[/tex]:
[tex]\[ 4^4 \times 4^2 = 4^{4+2} = 4^6 = 4096 \][/tex]
3. [tex]\( 3^2 \times 3^2 \)[/tex]:
[tex]\[ 3^2 \times 3^2 = 3^{2+2} = 3^4 = 81 \][/tex]
### 8. Dividing Powers with the Same Base (again)
1. [tex]\( 10^6 \div 10^4 \)[/tex]:
[tex]\[ 10^6 \div 10^4 = 10^{6-4} = 10^2 = 100 \, (\text{or } 100.0) \][/tex]
2. [tex]\( 8^3 \div 8^2 \)[/tex]:
[tex]\[ 8^3 \div 8^2 = 8^{3-2} = 8^1 = 8 \, (\text{or } 8.0) \][/tex]
3. [tex]\( 7^6 \div 7^3 \)[/tex]:
[tex]\[ 7^6 \div 7^3 = 7^{6-3} = 7^3 = 343 \, (\text{or } 343.0) \][/tex]
### 9. Multiplying Powers with the Same Base (again)
1. [tex]\( 5^3 \times 5^2 \)[/tex]:
[tex]\[ 5^3 \times 5^2 = 5^{3+2} = 5^5 = 3125 \][/tex]
2. [tex]\( 10^3 \times 10^4 \)[/tex]:
[tex]\[ 10^3 \times 10^4 = 10^{3+4} = 10^7 = 10000000 \][/tex]
3. [tex]\( 15^2 \times 15^1 \)[/tex]:
[tex]\[ 15^2 \times 15^1 = 15^{2+1} = 15^3 = 3375 \][/tex]
### 10. Dividing Powers with the Same Base (again)
1. [tex]\( 2^8 \div 2^3 \)[/tex]:
[tex]\[ 2^8 \div 2^3 = 2^{8-3} = 2^5 = 32 \, (\text{or } 32.0) \][/tex]
2. [tex]\( 3^9 \div 3^7 \)[/tex]:
[tex]\[ 3^9 \div 3^7 = 3^{9-7} = 3^2 = 9 \, (\text{or } 9.0) \][/tex]
3. [tex]\( 6^6 \div 6^3 \)[/tex]:
[tex]\[ 6^6 \div 6^3 = 6^{6-3} = 6^3 = 216 \, (\text{or } 216.0) \][/tex]
So, the detailed results are:
[tex]\[ 3^3 \times 3^3 = 729, \quad 2^2 \times 2^2 = 16, \quad 7^4 \div 7^2 = 49, \quad 9^5 \div 9^3 = 81, \quad 16^4 \div 16^2 = 256, \][/tex]
[tex]\[ 6^4 \times 6^1 = 7776, \quad 4^4 \times 4^2 = 4096, \quad 3^2 \times 3^2 = 81, \][/tex]
[tex]\[ 10^6 \div 10^4 = 100, \quad 8^3 \div 8^2 = 8, \quad 7^6 \div 7^3 = 343, \][/tex]
[tex]\[ 5^3 \times 5^2 = 3125, \quad 10^3 \times 10^4 = 10000000, \quad 15^2 \times 15^1 = 3375, \][/tex]
[tex]\[ 2^8 \div 2^3 = 32, \quad 3^9 \div 3^7 = 9, \quad 6^6 \div 6^3 = 216. \][/tex]
These results align perfectly with the desired answers for each of the given calculations.
### 5. Multiplying Powers with the Same Base
When multiplying powers with the same base, we add the exponents according to the rule [tex]\(a^m \times a^n = a^{m+n}\)[/tex]:
1. [tex]\( 3^3 \times 3^3 \)[/tex]:
[tex]\[ 3^3 \times 3^3 = 3^{3+3} = 3^6 = 729 \][/tex]
2. [tex]\( 2^2 \times 2^2 \)[/tex]:
[tex]\[ 2^2 \times 2^2 = 2^{2+2} = 2^4 = 16 \][/tex]
### 6. Dividing Powers with the Same Base
When dividing powers with the same base, we subtract the exponents according to the rule [tex]\(a^m \div a^n = a^{m-n}\)[/tex]:
1. [tex]\( 7^4 \div 7^2 \)[/tex]:
[tex]\[ 7^4 \div 7^2 = 7^{4-2} = 7^2 = 49 \, (\text{or } 49.0) \][/tex]
2. [tex]\( 9^5 \div 9^3 \)[/tex]:
[tex]\[ 9^5 \div 9^3 = 9^{5-3} = 9^2 = 81 \, (\text{or } 81.0) \][/tex]
3. [tex]\( 16^4 \div 16^2 \)[/tex]:
[tex]\[ 16^4 \div 16^2 = 16^{4-2} = 16^2 = 256 \, (\text{or } 256.0) \][/tex]
### 7. Multiplying Powers with the Same Base (again)
1. [tex]\( 6^4 \times 6^1 \)[/tex]:
[tex]\[ 6^4 \times 6^1 = 6^{4+1} = 6^5 = 7776 \][/tex]
2. [tex]\( 4^4 \times 4^2 \)[/tex]:
[tex]\[ 4^4 \times 4^2 = 4^{4+2} = 4^6 = 4096 \][/tex]
3. [tex]\( 3^2 \times 3^2 \)[/tex]:
[tex]\[ 3^2 \times 3^2 = 3^{2+2} = 3^4 = 81 \][/tex]
### 8. Dividing Powers with the Same Base (again)
1. [tex]\( 10^6 \div 10^4 \)[/tex]:
[tex]\[ 10^6 \div 10^4 = 10^{6-4} = 10^2 = 100 \, (\text{or } 100.0) \][/tex]
2. [tex]\( 8^3 \div 8^2 \)[/tex]:
[tex]\[ 8^3 \div 8^2 = 8^{3-2} = 8^1 = 8 \, (\text{or } 8.0) \][/tex]
3. [tex]\( 7^6 \div 7^3 \)[/tex]:
[tex]\[ 7^6 \div 7^3 = 7^{6-3} = 7^3 = 343 \, (\text{or } 343.0) \][/tex]
### 9. Multiplying Powers with the Same Base (again)
1. [tex]\( 5^3 \times 5^2 \)[/tex]:
[tex]\[ 5^3 \times 5^2 = 5^{3+2} = 5^5 = 3125 \][/tex]
2. [tex]\( 10^3 \times 10^4 \)[/tex]:
[tex]\[ 10^3 \times 10^4 = 10^{3+4} = 10^7 = 10000000 \][/tex]
3. [tex]\( 15^2 \times 15^1 \)[/tex]:
[tex]\[ 15^2 \times 15^1 = 15^{2+1} = 15^3 = 3375 \][/tex]
### 10. Dividing Powers with the Same Base (again)
1. [tex]\( 2^8 \div 2^3 \)[/tex]:
[tex]\[ 2^8 \div 2^3 = 2^{8-3} = 2^5 = 32 \, (\text{or } 32.0) \][/tex]
2. [tex]\( 3^9 \div 3^7 \)[/tex]:
[tex]\[ 3^9 \div 3^7 = 3^{9-7} = 3^2 = 9 \, (\text{or } 9.0) \][/tex]
3. [tex]\( 6^6 \div 6^3 \)[/tex]:
[tex]\[ 6^6 \div 6^3 = 6^{6-3} = 6^3 = 216 \, (\text{or } 216.0) \][/tex]
So, the detailed results are:
[tex]\[ 3^3 \times 3^3 = 729, \quad 2^2 \times 2^2 = 16, \quad 7^4 \div 7^2 = 49, \quad 9^5 \div 9^3 = 81, \quad 16^4 \div 16^2 = 256, \][/tex]
[tex]\[ 6^4 \times 6^1 = 7776, \quad 4^4 \times 4^2 = 4096, \quad 3^2 \times 3^2 = 81, \][/tex]
[tex]\[ 10^6 \div 10^4 = 100, \quad 8^3 \div 8^2 = 8, \quad 7^6 \div 7^3 = 343, \][/tex]
[tex]\[ 5^3 \times 5^2 = 3125, \quad 10^3 \times 10^4 = 10000000, \quad 15^2 \times 15^1 = 3375, \][/tex]
[tex]\[ 2^8 \div 2^3 = 32, \quad 3^9 \div 3^7 = 9, \quad 6^6 \div 6^3 = 216. \][/tex]
These results align perfectly with the desired answers for each of the given calculations.
