- Identify the pattern where each term is the previous term divided by 7 as the exponent decreases.
- Apply the negative exponent rule \(a^{-n} = \frac{1}{a^n}\) to find that \(7^{-5} = \frac{1}{7^5}\).
- Calculate \(7^5 = 16807\) to determine that \(7^{-5} = \frac{1}{16807}\).
- Conclude that Ramesh's claim is incorrect because powers of a positive number are always positive.
- Final answer: \(\boxed{\text{Ramesh is not correct}}\)
Explanation
- Analyzing the Pattern of Powers of 7
The provided table shows the pattern of powers of 7 for both positive and zero exponents. The sequence is \(7^4 = 2401\), \(7^3 = 343\), \(7^2 = 49\), \(7^1 = 7\), \(7^0 = 1\). Observing the pattern as we decrease the exponent by 1, the value is divided by 7. Following this logic, we continue the sequence for negative exponents: \(7^{-1} = \frac{1}{7}\). Continuing further, \(7^{-2} = \frac{1}{7} \div 7 = \frac{1}{49}\), \(7^{-3} = \frac{1}{343}\), \(7^{-4} = \frac{1}{2401}\), and \(7^{-5} = \frac{1}{16807}\).
- Applying the Law of Exponents
The definition of negative exponents states that for any non-zero base \(a\) and integer \(n\), the expression \(a^{-n}\) is equal to \(\frac{1}{a^n}\). Applying this to the given problem:
\[7^{-5} = \frac{1}{7^5}\]
First, we calculate \(7^5\):
\[7^5 = 7 \times 7 \times 7 \times 7 \times 7 = 16807\]
Therefore,
\[7^{-5} = \frac{1}{16807}\]
This confirms that the value is a very small positive fraction, not a negative integer.
- Evaluating Ramesh's Claim
Ramesh claimed that \(7^{-5} = -16,807\). Based on the pattern of division and the rule of negative exponents, this claim is incorrect. Ramesh likely confused the negative exponent sign with the resulting value being negative. However, powers of a positive base (like 7) are always positive regardless of the exponent. Thus, \(7^{-5}\) must be \(\frac{1}{16807}\). The correct statement is: Ramesh is not correct because as the exponents decrease, the previous value is divided by 7, so \(7^{-5} = 1 \div 7 \div 7 \div 7 \div 7 \div 7 = \frac{1}{16807}\).
Examples
Understanding negative exponents is essential in fields like scientific notation, where extremely small numbers are represented. For example, a scientist measuring the wavelength of a specific light might use scientific notation like \(5 \times 10^{-7}\) meters. By understanding that \(10^{-7} = \frac{1}{10,000,000}\), they can grasp the magnitude of the measurement. Just like in the case of 7, recognizing that negative exponents lead to small fractions rather than negative numbers helps in correctly interpreting data across various scientific and technical disciplines.