Question

Simplify: 100m^12n^4, m >= 0 n >= 0
a)m ^ 3 * n * sqrt(10) m >= 0 n >= 0
b)25m³ n, m >= 0 n >= 0
c)100m2 n, m >= 0 n >= 0
d)100m³n, m >= 0 n >= 0

Answer 1

To simplify the expression \( 100m^{12}n^{4} \), we can break it down as follows:

1. **Identify the factors**: The 100 can be expressed as \( 10^2 \).
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:

\[
100m^{12}n^{4} = 10^2 \cdot (m^6)^2 \cdot (n^2)^2
\]

3. **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:

1. **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.
2. **Option b)**: \( 25m^3 n \) - this has too low degrees of \( m \) and does not take into account the 100 factor.
3. **Option c)**: \( 100m^2 n \) - while it does have a factor of 100, \( m \) is too low here.
4. **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) \( 100m^3 n \), \( m \geq 0, n \geq 0 \)**.