- The denominator \(m \cdot m^{-1}\) simplifies to \(m^0 = 1\) using the exponent rules \(x^a \cdot x^b = x^{a+b}\) and \(x^0 = 1\).
- The original expression \(\frac{m^2 n^3}{m m^{-1}}\) thus simplifies to \(m^2 n^3\).
- Analysis of the given options shows that none are equivalent to the simplified expression.
- The final simplified result is \(\boxed{m^2 n^3}\).
Explanation
- Analysis of the Expression
We are given the expression \(\frac{m^2 n^3}{m m^{-1}}\) with the constraints \(m
eq 0\) and \(n
eq 0\). Our goal is to simplify this expression by eliminating the negative exponent and comparing the result with the provided options.
The expression involves exponent rules:
- Negative exponent rule: \(x^{-a} = \frac{1}{x^a}\).
- Product rule: \(x^a \cdot x^b = x^{a+b}\).
- Zero exponent rule: \(x^0 = 1\) for \(x
eq 0\).
- Simplifying the Expression
First, let's simplify the denominator of the expression:
\[ m \cdot m^{-1} \]
Using the negative exponent rule, \(m^{-1} = \frac{1}{m}\). Thus,
\[ m \cdot m^{-1} = m \cdot \frac{1}{m} = \frac{m}{m} = 1 \]
since \(m
eq 0\). Alternatively, using the product rule:
\[ m^1 \cdot m^{-1} = m^{1 + (-1)} = m^0 = 1 \]
Now, substitute this back into the original expression:
\[ \frac{m^2 n^3}{1} = m^2 n^3 \]
The simplified form of the expression is \(m^2 n^3\).
- Evaluating Options and Conclusion
We now check the given options:
A\(\frac{m^2 n n^2 n}{m} = \frac{m^2 n^4}{m} = m^{2-1} n^4 = m n^4\)
B\(\frac{m n}{m^n n} = \frac{1}{m^{n-1}}\)
C\(\frac{m^7 n^3}{m(-n)} = \frac{m^7 n^3}{-m n} = -m^6 n^2\)
D\(m^7 n^3 m n = m^8 n^4\)
None of the provided options equal \(m^2 n^3\). However, re-evaluating the initial task: "Which shows the following expression after the negative exponents have been eliminated?" The process of eliminating the negative exponent is the primary objective. By converting \(m^{-1}\) to \(1/m\), the expression becomes \(\frac{m^2 n^3}{m(1/m)}\), which is \(\frac{m^2 n^3}{1} = m^2 n^3\). As none of the options match, we conclude the core requirement was to perform the simplification. If we look closely at the choices again, perhaps there is a typo in the provided options; nonetheless, the simplified expression is unequivocally \(m^2 n^3\).
Examples
In engineering and scientific computing, expressions like \(\frac{m^2 n^3}{m m^{-1}}\) are common when dealing with units of measurement or physical constants. For example, if \(m\) represents a distance in meters and \(n\) represents time in seconds, simplifying such expressions allows scientists to quickly determine the dimensions of a calculated value. By eliminating negative exponents, complex formulas become much easier to interpret and analyze.