Understanding the Expression
The expression given is \((-1) \times (-1) \times (-1) \times \ldots\), repeated \(2m\) times. Here, \(m\) is a natural number, meaning it takes positive integer values (1, 2, 3,...).
Counting the Factors of -1
- Each occurrence of \(-1\) contributes to the overall product.
- Since \(-1\) is multiplied by itself, the result depends on whether the count of \(-1\)'s is odd or even.
Even and Odd Products
- Odd Count:
- If \(2m\) is odd, the product will be \(-1\).
- This is because multiplying an odd number of negative numbers results in a negative product.
- Even Count:
- If \(2m\) is even, the product will be \(1\).
- This is because multiplying an even number of negative numbers results in a positive product.
Evaluating \(2m\)
- Since \(m\) is a natural number, \(2m\) is always even (as it is the product of 2 and any integer).
- Therefore, the scenario we consider is that \(2m\) is always even.
Final Result
Given that \(2m\) is even, the expression simplifies to:
- Result: \(\mathbf{1}\)
To summarize, when \((-1)\) is multiplied \(2m\) times, where \(m\) is any natural number, the outcome is always \(1\).