Let's address the problem step-by-step to clearly understand Reese’s error and the correct approach to simplify the expression.
Consider the original expression:
[tex]\[
\frac{10 m^3 n^5}{5 m n^2}
\][/tex]
1. Simplify the Coefficients:
[tex]\[
\frac{10}{5} = 2
\][/tex]
2. Simplify the Exponents for \(m\):
[tex]\[
\frac{m^3}{m} = m^{3-1} = m^2
\][/tex]
3. Simplify the Exponents for \(n\):
[tex]\[
\frac{n^5}{n^2} = n^{5-2} = n^3
\][/tex]
Putting it all together, the simplified expression is:
[tex]\[
2m^2n^3
\][/tex]
Reese's Error:
Reese attempted to combine \(m\) and \(n\) into \((mn)\) and then used the law of exponents on these combined bases incorrectly. He wrote:
[tex]\[
\frac{10(mn)^8}{5(mn)^2}
\][/tex]
By doing so, he misinterpreted the original problem. \(m\) and \(n\) are different variables and should be treated separately concerning their exponents and the rules of division.
Reese’s final answer:
[tex]\[
2(mn)^6
\][/tex]
is incorrect, as he incorrectly combined the exponents and the variables.
Correct Result: Based on the correct step-by-step simplification:
[tex]\[
2m^2n^3
\][/tex]
Reason for Reese's Error:
He used a law of exponents on factors with different bases which is not correct. He should have treated \(m\) and \(n\) separately in the exponentiation and division process to reach the final and correct simplified result:
[tex]\[
2m^2n^3
\][/tex]
In summary, Reese should have divided the exponents correctly, treating the variables [tex]\(m\)[/tex] and [tex]\(n\)[/tex] separately, to arrive at the final answer of [tex]\(\mathbf{2m^2n^3}\)[/tex].
