To simplify the expression \( 100m^{12}n^{4} \), we can break it down as follows:
- Identify the factors: The 100 can be expressed as \( 10^2 \).
- Look for perfect squares: This means we want to express \( m^{12} \) and \( n^{4} \) in terms of squares because we might want to factor that out for a square root.
Now we can express the terms under a square root:
\[ 100m^{12}n^{4} = 10^2 \cdot (m^6)^2 \cdot (n^2)^2 \]
- Express as a whole square: Thus, we can write:
\[ \sqrt{100m^{12}n^{4}} = \sqrt{10^2} \cdot \sqrt{(m^6)^2} \cdot \sqrt{(n^2)^2} = 10 \cdot m^6 \cdot n^{2} \]
Since \( m \) and \( n \) are non-negative, we can simplify to get the overall expression:
\[ \sqrt{100m^{12}n^{4}} = 10m^6n^{2} \]
Since we need to determine which of the provided options best simplifies \( 100m^{12}n^{4} \):
Let's consider the options:
a) \( m^3 n \sqrt{10} \) b) \( 25m^3 n \) c) \( 100m^2 n \) d) \( 100m^3 n \)
To find relationships between \( 100m^{12}n^{4} \) and these options, we can calculate each option:
- Option a): \( m^3 n \sqrt{10} \) - does not match since we calculated and don't have a square root, plus the degrees of \( m \) and \( n \) don't add up.
- Option b): \( 25m^3 n \) - this has too low degrees of \( m \) and does not take into account the 100 factor.
- Option c): \( 100m^2 n \) - while it does have a factor of 100, \( m \) is too low here.
- Option d): \( 100m^3 n \) - this is the only one with the proper factorization of \( 100 \) and maintains non-negative powers of \( m \) and \( n \).
Thus, based on the simplification above, the answer is:
d) \( 100m^3 n \), \( m \geq 0, n \geq 0 \).
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