To simplify the quotient [tex]\(\frac{2 m^9 n^4}{-4 m^{-3} n^{-2}}\)[/tex], let's break it down step by step.
1. Simplify the coefficients:
The coefficients are [tex]\(2\)[/tex] in the numerator and [tex]\(-4\)[/tex] in the denominator.
[tex]\[
\frac{2}{-4} = -0.5
\][/tex]
2. Combine the [tex]\(m\)[/tex] terms using the properties of exponents:
When dividing like bases, subtract the exponents.
For the [tex]\(m\)[/tex] terms, we have [tex]\(m^9\)[/tex] in the numerator and [tex]\(m^{-3}\)[/tex] in the denominator.
[tex]\[
\frac{m^9}{m^{-3}} = m^{9 - (-3)} = m^{9 + 3} = m^{12}
\][/tex]
3. Combine the [tex]\(n\)[/tex] terms using the properties of exponents:
Similarly, for the [tex]\(n\)[/tex] terms, we have [tex]\(n^4\)[/tex] in the numerator and [tex]\(n^{-2}\)[/tex] in the denominator.
[tex]\[
\frac{n^4}{n^{-2}} = n^{4 - (-2)} = n^{4 + 2} = n^6
\][/tex]
4. Combine the simplified terms:
After simplifying the coefficients and the terms with [tex]\(m\)[/tex] and [tex]\(n\)[/tex], we get:
[tex]\[
-0.5 \cdot m^{12} \cdot n^6
\][/tex]
5. Write the final simplified quotient:
The quotient in simplest form is:
[tex]\[
-0.5 m^{12} n^6
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
-\frac{m^{12} n^6}{2}
\][/tex]
