Verified Answer :
- Simplify multiplication by adding exponents: \(a^m \cdot a^n = a^{m+n}\).
- Simplify division by subtracting exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
- Treat lone bases as having an exponent of 1, e.g., \(7 = 7^1\).
- Final results for the provided expressions: 1. \(4^5\), 2. \(9^{13}\), 6. \(5^0\), 7. \(4^6\), 8. \(7^3\), 11. \(3^2\), 12. \(11^5\), 16. \(10^4\), 17. \(3^4\), 21. \(7^2\), 22. \(10^0\), 26. \((-s)^8\), 27. \(2^{16}\).
Explanation
- Analysis of Exponent Laws
The objective is to simplify each exponential expression into a single base raised to a single integer exponent \(b^k\). We rely on the fundamental laws of exponents:
- Product rule: \(a^m \cdot a^n = a^{m+n}\).
- Quotient rule: \(\frac{a^m}{a^n} = a^{m-n}\).
- Single exponent rule: \(a = a^1\).
- Power rule for products: \(a^m \cdot a^n \cdot a^p = a^{m+n+p}\).
We will apply these systematically to each expression.
- Calculation and Simplification of Expressions
- \(4^2 \cdot 4^3 = 4^{2+3} = 4^5\)
- \(9^9 \cdot 9^4 = 9^{9+4} = 9^{13}\)
- \(5^{-4} \cdot 5^2 \cdot 5^2 = 5^{-4+2+2} = 5^0\)
- \(\frac{4^9}{4^3} = 4^{9-3} = 4^6\)
- \(\frac{7^4}{7^1} = 7^{4-1} = 7^3\)
- \(\frac{3^7 \cdot 3^3}{3^8} = \frac{3^{10}}{3^8} = 3^{10-8} = 3^2\)
- \(\frac{11^9 \cdot 11^3}{11^2 \cdot 11^5} = \frac{11^{12}}{11^7} = 11^{12-7} = 11^5\)
- \(\frac{10^5 \cdot 10^{-3}}{10^{-5} \cdot 10^3} = \frac{10^2}{10^{-2}} = 10^{2-(-2)} = 10^4\)
- \(\frac{3^6 \cdot 3^3}{3^5} = \frac{3^9}{3^5} = 3^{9-5} = 3^4\)
- \(\frac{7^8}{7^5 \cdot 7^1} = \frac{7^8}{7^6} = 7^{8-6} = 7^2\)
- \(\frac{10^6}{10^3 \cdot 10^3} = \frac{10^6}{10^6} = 10^0\)
- \((-s)^2(-s)^3(-s)^3 = (-s)^{2+3+3} = (-s)^8\)
- \(\frac{2^{11} \cdot 2^8}{2^3} = \frac{2^{19}}{2^3} = 2^{16}\)
- Summary of Results
The expressions have been simplified successfully according to the rules of exponents, resulting in the following forms:
- \(4^5\), 2. \(9^{13}\), 6. \(5^0\), 7. \(4^6\), 8. \(7^3\), 11. \(3^2\), 12. \(11^5\), 16. \(10^4\), 17. \(3^4\), 21. \(7^2\), 22. \(10^0\), 26. \((-s)^8\), 27. \(2^{16}\).
Examples
Exponent rules are essential in scientific fields, particularly when dealing with extremely large or small numbers. For example, in computer science, memory is measured in bits, and transferring data between systems requires calculating total capacity using powers of two. If a file is divided into \(2^{11}\) chunks of \(2^8\) bits each, the total size is \(2^{11} \cdot 2^8 = 2^{19}\) bits. Understanding these rules allows for efficient data handling and quick calculations without needing to multiply the entire numbers.