To represent the number 200 in the form \( 2^n \cdot 5^m \), we can start by factoring 200 into its prime factors.
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First, let's find the prime factorization of 200: \[ 200 = 2 \times 100 \] \[ 100 = 2 \times 50 \] \[ 50 = 2 \times 25 \] \[ 25 = 5 \times 5 \]
So, combining these results: \[ 200 = 2^3 \times 5^2 \]
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In this case, we have:
- \( n = 3 \) (the power of 2),
- \( m = 2 \) (the power of 5).
Thus, we can express 200 in the form \( 2^n \cdot 5^m \) as: \[ 200 = 2^3 \cdot 5^2 \]