- The expression \(\frac{a^m}{a^n} = a^{m-n}\) for \(a
eq 0\) is a standard, correct law of exponents known as the quotient rule.
- The expression \((a^m)^n = a^{m+n}\) is incorrect; the correct power of a power rule is \((a^m)^n = a^{m \cdot n}\).
- Since \((a^m)^n = a^{m+n}\) fails for general values of \(a, m, n\), it is not a valid law of exponents.
- The final answer is \(\boxed{\text{A}}\).
Explanation
- Analysis of the Provided Options
We are given two expressions involving exponents and need to determine which one is not a standard law of exponents. The two expressions are:
- \( (a^m)^n = a^{m+n} \)
- \( \frac{a^m}{a^n} = a^{m-n} \) (where \( a
eq 0 \))
An exponent law is an identity that holds true for all valid real numbers \( a \) and all integers \( m \) and \( n \). We evaluate each based on fundamental algebraic definitions.
- Evaluating Option B
Consider the second option, \( \frac{a^m}{a^n} = a^{m-n} \) for \( a
eq 0 \). This is one of the fundamental laws of exponents, often referred to as the quotient rule. It is derived from the fact that \( \frac{a^m}{a^n} = a^m \cdot a^{-n} = a^{m+(-n)} = a^{m-n} \). Thus, this expression is a valid law of exponents.
- Evaluating Option A
Consider the first option, \( (a^m)^n = a^{m+n} \). According to the power of a power rule, the correct identity is \( (a^m)^n = a^{m \cdot n} \). Let us test this with a simple example where \( a=2, m=2, n=3 \).
For the left side: \( (2^2)^3 = 4^3 = 64 \).
For the right side (using the proposed rule \( a^{m+n} \)): \( 2^{2+3} = 2^5 = 32 \).
Since \( 64
eq 32 \), the proposed rule \( (a^m)^n = a^{m+n} \) is incorrect. Therefore, it is not a valid law of exponents.
- Conclusion
We have verified that the quotient rule is a valid law of exponents, whereas the power rule provided in Option A contradicts the established law \( (a^m)^n = a^{m \cdot n} \). Consequently, option A is not a law of exponents. The final answer is option A.
Examples
Understanding exponent laws is crucial for simplifying complex expressions in science and finance. For instance, if you are calculating compound interest where an investment grows by a factor each year, you use exponents to track the total growth over time. Knowing that \((a^m)^n = a^{m \cdot n}\) allows you to calculate the result of interest being applied annually and then scaled over a multi-year period quickly, avoiding tedious step-by-step multiplication. Mastering these rules ensures that you can manage data or finances efficiently.