- Identify the expression as \(\frac{m^7 n^3}{m n^{-1}}\) and the rule \(\frac{1}{x^{-a}} = x^a\).
- Substitute \(n^{-1}\) in the denominator to bring it to the numerator as \(n^1\), or rewrite the expression as \(\frac{m^7 n^3}{m/n}\).
- Simplify the fraction to \(\frac{m^7 n^3 n}{m}\).
- Identify that this matches option A. Final answer: \(\boxed{A}\).
Explanation
- Analyzing the Problem and Rules of Exponents
The goal is to simplify the expression \(\frac{m^7 n^3}{m n^{-1}}\) by eliminating the negative exponent. We are given that \(m
eq 0\) and \(n
eq 0\). Negative exponents can be managed using the rule that for any non-zero base \(x\), \(x^{-a} = \frac{1}{x^a}\). Conversely, this means that \(\frac{1}{x^{-a}} = x^a\). In our expression, we have \(n^{-1}\) in the denominator, which is equivalent to \(\frac{1}{n}\).
- Simplifying the Expression
To eliminate the negative exponent, we substitute \(n^{-1}\) with \(\frac{1}{n}\) in the expression:
\[\frac{m^7 n^3}{m n^{-1}} = \frac{m^7 n^3}{m \left(\frac{1}{n}\right)}\]
Now, the expression is in the form of a complex fraction. To simplify \(\frac{m^7 n^3}{m/n}\), we multiply by the reciprocal of the denominator. Dividing by a fraction is the same as multiplying by its reciprocal:
\[\frac{m^7 n^3}{m/n} = m^7 n^3 \cdot \frac{n}{m} = \frac{m^7 n^3 n}{m}\]
Alternatively, we can use the property \(\frac{1}{n^{-1}} = n^1\). Since \(n^{-1}\) is in the denominator, its reciprocal, \(n^1\), moves to the numerator:
\[\frac{m^7 n^3}{m n^{-1}} = \frac{m^7 n^3 n^1}{m} = \frac{m^7 n^3 n}{m}\]
- Conclusion and Selection of the Correct Option
Comparing our simplified result, \(\frac{m^7 n^3 n}{m}\), with the given options:
A\(\frac{m^7 n^3 n}{m}\)
B\(m^7 n^3 m n\)
C\(\frac{m^7 n^3}{m(-n)}\)
D\(\frac{m n}{m^7 n^3}\)
The derived expression matches option A. Thus, the expression \(\frac{m^7 n^3}{m n^{-1}}\) with the negative exponent eliminated is \(\frac{m^7 n^3 n}{m}\).
Examples
In data processing, you might deal with very small or very large quantities represented as powers of base-10 or other variables. For instance, if you are scaling a variable by a factor like \(n^{-1}\), which is \(\frac{1}{n}\), in a formula for signal processing or scientific computation, simplifying the expression to remove negative exponents makes it easier to implement in programming languages. By turning \(\frac{m^7 n^3}{m n^{-1}}\) into \(\frac{m^7 n^3 n}{m}\), you convert a division into a multiplication, which is often faster and more precise in digital calculations.