To determine whether Tyrone or Alisha is correct, let's analyze the expression step-by-step:
The original expression is:
\[ \left( \frac{m^2 n^{-3}}{m^6 n^{-1}} \right)^{3} \]
Tyrone's Claim
Tyrone suggests that the first step is to raise the numerator and denominator to the third power. If we follow this approach, it would look like this:
\[ \frac{(m^2 n^{-3})^3}{(m^6 n^{-1})^3} \]
This is indeed a valid step, but it is typically more efficient to simplify the expression first before raising it to a power.
Alisha's Claim
Alisha suggests applying the quotient of powers first. The quotient of powers rule states that when dividing like bases, you subtract the exponents. So we can simplify the expression inside the parentheses first.
Let's simplify:
- Simplify the base before cubing:
- For \(m^2\) (numerator) and \(m^6\) (denominator): \[ \frac{m^2}{m^6} = m^{2-6} = m^{-4} \]
- For \(n^{-3}\) (numerator) and \(n^{-1}\) (denominator): \[ \frac{n^{-3}}{n^{-1}} = n^{-3 - (-1)} = n^{-3 + 1} = n^{-2} \]
So combining these, we have:
\[ \frac{m^2 n^{-3}}{m^6 n^{-1}} = m^{-4} n^{-2} \]
Now, we take this simplified expression and raise it to the third power:
\[ (m^{-4} n^{-2})^{3} = m^{-12} n^{-6} \]
Conclusion
Alisha's claim is correct. It is generally simpler and more efficient to first apply the quotient of powers before raising the entire expression to a power. In this way, we reduce the complexity of the expression, making it easier to work with.