Answer :
Let's solve each of these expressions step-by-step and convert them into simplified exponential form:
1. [tex]\(2^6 \times 2^4\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[2^6 \times 2^4 = 2^{6+4} = 2^{10}\][/tex]
2. [tex]\(3^2 \times 3^3 \times 3^4\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[3^2 \times 3^3 \times 3^4 = 3^{2+3+4} = 3^9\][/tex]
3. [tex]\(8^8 \div 8^7\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[8^8 \div 8^7 = 8^{8-7} = 8^1 = 8\][/tex]
4. [tex]\(x^{11} \times x^{11}\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[x^{11} \times x^{11} = x^{11+11} = x^{22}\][/tex]
5. [tex]\(12^x \times 12^9\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[12^x \times 12^9 = 12^{x+9}\][/tex]
6. [tex]\(\left(13^2\right)^4\)[/tex]
When raising an exponent to another power, you multiply the exponents:
[tex]\[\left(13^2\right)^4 = 13^{2 \times 4} = 13^8\][/tex]
7. [tex]\((-6)^7 \div (-6)^5\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[(-6)^7 \div (-6)^5 = (-6)^{7-5} = (-6)^2\][/tex]
8. [tex]\(\left(\frac{5}{7}\right)^3 \times \left(\frac{5}{7}\right)^3\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[\left(\frac{5}{7}\right)^3 \times \left(\frac{5}{7}\right)^3 = \left(\frac{5}{7}\right)^{3+3} = \left(\frac{5}{7}\right)^6\][/tex]
9. [tex]\(a^{20} \div a^8\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[a^{20} \div a^8 = a^{20-8} = a^{12}\][/tex]
10. [tex]\(\left[\left(\frac{-1}{2}\right)^3\right]^7\)[/tex]
When raising an exponent to another power, you multiply the exponents:
[tex]\[\left[\left(\frac{-1}{2}\right)^3\right]^7 = \left(\frac{-1}{2}\right)^{3 \times 7} = \left(\frac{-1}{2}\right)^{21}\][/tex]
11. [tex]\(a^3 \times b^3\)[/tex]
This expression does not reduce further involving powers with the same base, but it can be written as:
[tex]\[(a \times b)^3\][/tex]
12. [tex]\(\frac{(-6)^4}{7^4}\)[/tex]
When dividing exponents with different bases, you cannot simplify further, but it can be written as a single fraction:
[tex]\[\frac{(-6)^4}{7^4} = \left(\frac{-6}{7}\right)^4\][/tex]
13. [tex]\((-2)^{12} \times (-2)^{-6}\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[(-2)^{12} \times (-2)^{-6} = (-2)^{12 + (-6)} = (-2)^6\][/tex]
14. [tex]\(7^8 \div 7^{-7}\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[7^8 \div 7^{-7} = 7^{8-(-7)} = 7^{8+7} = 7^{15}\][/tex]
15. [tex]\(\left(4^4 \times 4^5\right) \div 4^7\)[/tex]
First, simplify the multiplication inside the parentheses by adding exponents:
[tex]\[4^4 \times 4^5 = 4^{4+5} = 4^9\][/tex]
Then, divide by [tex]\(4^7\)[/tex] by subtracting the exponents:
[tex]\[4^9 \div 4^7 = 4^{9-7} = 4^2\][/tex]
In summary:
1. [tex]\(2^{10}\)[/tex]
2. [tex]\(3^9\)[/tex]
3. [tex]\(8^1\)[/tex]
4. [tex]\(x^{22}\)[/tex]
5. [tex]\(12^{x+9}\)[/tex]
6. [tex]\(13^8\)[/tex]
7. [tex]\((-6)^2\)[/tex]
8. [tex]\(\left(\frac{5}{7}\right)^6\)[/tex]
9. [tex]\(a^{12}\)[/tex]
10. [tex]\(\left(\frac{-1}{2}\right)^{21}\)[/tex]
11. [tex]\((a \times b)^3\)[/tex]
12. [tex]\(\left(\frac{-6}{7}\right)^4\)[/tex]
13. [tex]\((-2)^6\)[/tex]
14. [tex]\(7^{15}\)[/tex]
15. [tex]\(4^2\)[/tex]
1. [tex]\(2^6 \times 2^4\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[2^6 \times 2^4 = 2^{6+4} = 2^{10}\][/tex]
2. [tex]\(3^2 \times 3^3 \times 3^4\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[3^2 \times 3^3 \times 3^4 = 3^{2+3+4} = 3^9\][/tex]
3. [tex]\(8^8 \div 8^7\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[8^8 \div 8^7 = 8^{8-7} = 8^1 = 8\][/tex]
4. [tex]\(x^{11} \times x^{11}\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[x^{11} \times x^{11} = x^{11+11} = x^{22}\][/tex]
5. [tex]\(12^x \times 12^9\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[12^x \times 12^9 = 12^{x+9}\][/tex]
6. [tex]\(\left(13^2\right)^4\)[/tex]
When raising an exponent to another power, you multiply the exponents:
[tex]\[\left(13^2\right)^4 = 13^{2 \times 4} = 13^8\][/tex]
7. [tex]\((-6)^7 \div (-6)^5\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[(-6)^7 \div (-6)^5 = (-6)^{7-5} = (-6)^2\][/tex]
8. [tex]\(\left(\frac{5}{7}\right)^3 \times \left(\frac{5}{7}\right)^3\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[\left(\frac{5}{7}\right)^3 \times \left(\frac{5}{7}\right)^3 = \left(\frac{5}{7}\right)^{3+3} = \left(\frac{5}{7}\right)^6\][/tex]
9. [tex]\(a^{20} \div a^8\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[a^{20} \div a^8 = a^{20-8} = a^{12}\][/tex]
10. [tex]\(\left[\left(\frac{-1}{2}\right)^3\right]^7\)[/tex]
When raising an exponent to another power, you multiply the exponents:
[tex]\[\left[\left(\frac{-1}{2}\right)^3\right]^7 = \left(\frac{-1}{2}\right)^{3 \times 7} = \left(\frac{-1}{2}\right)^{21}\][/tex]
11. [tex]\(a^3 \times b^3\)[/tex]
This expression does not reduce further involving powers with the same base, but it can be written as:
[tex]\[(a \times b)^3\][/tex]
12. [tex]\(\frac{(-6)^4}{7^4}\)[/tex]
When dividing exponents with different bases, you cannot simplify further, but it can be written as a single fraction:
[tex]\[\frac{(-6)^4}{7^4} = \left(\frac{-6}{7}\right)^4\][/tex]
13. [tex]\((-2)^{12} \times (-2)^{-6}\)[/tex]
When multiplying exponents with the same base, you add the exponents:
[tex]\[(-2)^{12} \times (-2)^{-6} = (-2)^{12 + (-6)} = (-2)^6\][/tex]
14. [tex]\(7^8 \div 7^{-7}\)[/tex]
When dividing exponents with the same base, you subtract the exponents:
[tex]\[7^8 \div 7^{-7} = 7^{8-(-7)} = 7^{8+7} = 7^{15}\][/tex]
15. [tex]\(\left(4^4 \times 4^5\right) \div 4^7\)[/tex]
First, simplify the multiplication inside the parentheses by adding exponents:
[tex]\[4^4 \times 4^5 = 4^{4+5} = 4^9\][/tex]
Then, divide by [tex]\(4^7\)[/tex] by subtracting the exponents:
[tex]\[4^9 \div 4^7 = 4^{9-7} = 4^2\][/tex]
In summary:
1. [tex]\(2^{10}\)[/tex]
2. [tex]\(3^9\)[/tex]
3. [tex]\(8^1\)[/tex]
4. [tex]\(x^{22}\)[/tex]
5. [tex]\(12^{x+9}\)[/tex]
6. [tex]\(13^8\)[/tex]
7. [tex]\((-6)^2\)[/tex]
8. [tex]\(\left(\frac{5}{7}\right)^6\)[/tex]
9. [tex]\(a^{12}\)[/tex]
10. [tex]\(\left(\frac{-1}{2}\right)^{21}\)[/tex]
11. [tex]\((a \times b)^3\)[/tex]
12. [tex]\(\left(\frac{-6}{7}\right)^4\)[/tex]
13. [tex]\((-2)^6\)[/tex]
14. [tex]\(7^{15}\)[/tex]
15. [tex]\(4^2\)[/tex]
