- A negative exponent is defined as \(a^{-n} = \frac{1}{a^n}\), so \(7^{-5} = \frac{1}{7^5} = \frac{1}{16,807}\).
- Ramesh is incorrect because he confuses the value \(\frac{1}{16,807}\) with a negative integer.
- The first statement correctly explains that decreasing exponents imply dividing by the base repeatedly, confirming \(7^{-5} = \frac{1}{16,807}\).
- The second statement is incorrect because it contains a faulty calculation step (\(\frac{1}{7^4} \div 7 = \frac{1}{7^3}\)).
- Final Answer: The first statement is correct.
Explanation
- Analyzing the Definition of Negative Exponents
To evaluate Ramesh's claim that \(7^{-5} = -16,807\), we must apply the definition of negative exponents. The rule for any non-zero base \(a\) and integer exponent \(n\) is \(a^{-n} = \frac{1}{a^n}\). Applying this to the given expression, we have \(7^{-5} = \frac{1}{7^5}\). Calculate \(7^5\): \(7 \times 7 \times 7 \times 7 \times 7 = 49 \times 49 \times 7 = 2,401 \times 7 = 16,807\). Therefore, the actual value is \(\frac{1}{16,807}\).
- Evaluating the Explanatory Statements
Now we evaluate the two provided statements to see which correctly explains why Ramesh's claim is incorrect.
Statement 1 says: "Ramesh is not correct because as the exponents decrease, the previous value is divided by 7, so \(7^{-5} = 1 \div 7 \div 7 \div 7 \div 7 \div 7 = \frac{1}{16,807}\)." This matches our calculation perfectly, as \(7^{-5}\) is indeed obtained by starting at 1 and dividing by 7 five times.
Statement 2 says: "Ramesh is not correct because \(7^{-5}\) is equivalent to \(\frac{1}{7^5}\), which has the same value as \(\frac{1}{7^4} \div 7 = \frac{1}{7^3} = \frac{1}{343}\)." While the first part correctly states that \(7^{-5} = \frac{1}{7^5}\), the subsequent steps are mathematically flawed because \(\frac{1}{7^4} \div 7 = \frac{1}{7^5}\), not \(\frac{1}{7^3}\). Thus, Statement 2 contains a calculation error.
- Conclusion
Based on our analysis, Statement 1 is the correct explanation. It properly defines the process of negative exponentiation as repeated division by the base, leading to the correct value of \(\frac{1}{16,807}\). Statement 2 fails because it contains an incorrect simplification step. Therefore, the first statement is the valid one.
Examples
Understanding how exponents work helps in various scientific and financial calculations. For example, in exponential decay models (like calculating the half-life of a radioactive isotope or the depreciation of an asset over time), you often deal with negative exponents. Recognizing that \(a^{-n} = \frac{1}{a^n}\) ensures you calculate shrinking values correctly instead of confusing them with negative numbers, which is essential for accurate modeling.