Question

what is the quotient \\(\frac{2m^9n^4}{-4m^{-3}n^{-2}}\\) in simplest form? assume \\(m \
eq 0, n \
eq 0\\).\
\\(8m^{12}n^6\\)\
\\(-\frac{m^{12}n^6}{2}\\)\
\\(6m^{12}n^6\\)\
\\(-\frac{m^{27}n^8}{2}\\)

Explanation:

Step1: Simplify coefficients

Divide the coefficients: $\frac{2}{-4} = -\frac{1}{2}$.

Step2: Simplify \( m \)-terms

Use exponent rule \( \frac{a^m}{a^n} = a^{m - n} \) (wait, no, when dividing with negative exponents, it's \( a^m \div a^{-n} = a^{m + n} \)). So for \( m \): \( m^9 \div m^{-3} = m^{9 - (-3)} = m^{12} \).

Step3: Simplify \( n \)-terms

For \( n \): \( n^4 \div n^{-2} = n^{4 - (-2)} = n^6 \).

Step4: Combine results

Multiply the coefficient, \( m \)-term, and \( n \)-term: \( -\frac{1}{2} \times m^{12} \times n^6 = -\frac{m^{12}n^6}{2} \).

Answer:

\(-\frac{m^{12}n^6}{2}\) (corresponding to the option with this expression)