Question
What is the simplified form of \( \frac{-24 m^{5} n^{4}}{8 m^{-7} n^{-2}} ? \) Assume \( m \neq 0, n \neq 0 \)
Ask by Ruiz Frank. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form is \( -3 m^{12} n^{6} \).
Solution
\[
\frac{-24 m^{5} n^{4}}{8 m^{-7} n^{-2}} = \frac{-24}{8} \cdot \frac{m^{5}}{m^{-7}} \cdot \frac{n^{4}}{n^{-2}}
\]
\[
\frac{-24}{8} = -3
\]
\[
\frac{m^{5}}{m^{-7}} = m^{5 - (-7)} = m^{12}
\]
\[
\frac{n^{4}}{n^{-2}} = n^{4 - (-2)} = n^{6}
\]
\[
\text{Thus, the simplified form is } -3 m^{12} n^{6}.
\]
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Simplify this solution Beyond the Answer
To simplify \( \frac{-24 m^{5} n^{4}}{8 m^{-7} n^{-2}} \), first, we divide the coefficients: \(-24 \div 8 = -3\). Next, for the powers of \( m \), use the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \): \[ m^{5} \div m^{-7} = m^{5 - (-7)} = m^{5 + 7} = m^{12}. \] Now for the powers of \( n \): \[ n^{4} \div n^{-2} = n^{4 - (-2)} = n^{4 + 2} = n^{6}. \] Putting it all together, we have: \[ \frac{-24 m^{5} n^{4}}{8 m^{-7} n^{-2}} = -3 m^{12} n^{6}. \] Thus, the simplified form is \[ \boxed{-3 m^{12} n^{6}}. \]
