Answer :
Answer:
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Step-by-step explanation:
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Answer:
To determine whether \(3^{7n - 2}\) is rational, irrational, or natural, we need to understand the properties of the exponent and the base.
1. **Base**: The base is 3, which is an integer and a rational number.
2. **Exponent**: The exponent is \(7n - 2\). The value of \(7n - 2\) depends on the value of \(n\).
- If \(n\) is an integer, then \(7n - 2\) is also an integer (since the product and difference of integers are integers).
Now, let's analyze the nature of \(3^{7n - 2}\):
- **Rational vs. Irrational**: An expression of the form \(a^b\) is rational if \(a\) is a rational number and \(b\) is an integer. Since 3 is a rational number and \(7n - 2\) is an integer, \(3^{7n - 2}\) is a rational number.
- **Natural Number**: A natural number is a positive integer (1, 2, 3, ...). Whether \(3^{7n - 2}\) is a natural number depends on the value of \(n\). If \(7n - 2\) is a non-negative integer (i.e., \(7n - 2 \geq 0\)), then \(3^{7n - 2}\) is a natural number. However, this is not necessarily true for all \(n\).
Given that the expression \(3^{7n - 2}\) is rational and whether it is a natural number depends on \(n\):
The correct answer to describe the number \(3^{7n - 2}\) in general is:
- A rational number
Because this is always true regardless of the value of \(n\). The expression being a natural number is conditional on \(n\) and hence not always true.