Let's evaluate the expression [tex]\(\left(\frac{r s^{-2}}{r^2 s^{-3}}\right)^{-1}\)[/tex] when [tex]\(r = -1\)[/tex] and [tex]\(s = -2\)[/tex] by examining the steps given by Selena and Drake.
Selena's Work:
1. Expression: [tex]\(\left(\frac{r s^{-2}}{r^2 s^{-3}}\right)^{-1}\)[/tex]
2. Simplify the expression inside the parentheses:
[tex]\[
\frac{r s^{-2}}{r^2 s^{-3}}
\][/tex]
3. Divide the exponents:
[tex]\[
= \frac{r}{r^2} \cdot \frac{s^{-2}}{s^{-3}} = r^{-1} \cdot s^{1}
\][/tex]
4. Simplify:
[tex]\[
= \left(r^{-1} s\right)^{-1}
\][/tex]
5. Take the reciprocal of the simplified expression:
[tex]\[
= \frac{r}{s}
\][/tex]
6. Substitute [tex]\(r = -1\)[/tex] and [tex]\(s = -2\)[/tex]:
[tex]\[
= \frac{-1}{-2} = \frac{1}{2}
\][/tex]
Drake's Work:
1. Expression: [tex]\(\left(\frac{r s^{-2}}{r^2 s^{-3}}\right)^{-1}\)[/tex]
2. Substitute [tex]\(r = -1\)[/tex] and [tex]\(s = -2\)[/tex]:
[tex]\[
= \left(\frac{(-1)(-2)^{-2}}{(-1)^2(-2)^{-3}}\right)^{-1}
\][/tex]
3. Simplify the exponents:
[tex]\[
= \left(\frac{(-1)(\frac{1}{4})}{(1)(-\frac{1}{8})}\right)^{-1} = \left(\frac{-1 \cdot \frac{1}{4}}{\frac{1}{-8}}\right)^{-1}
\][/tex]
4. Simplify further:
[tex]\[
= \left(\frac{-\frac{1}{4}}{-\frac{1}{8}}\right)^{-1}
\][/tex]
5. Evaluate the fractions:
[tex]\[
= \left(\frac{-1}{4} \cdot \frac{8}{-1}\right)^{-1} = \left(2\right)^{-1} = \frac{1}{2}
\][/tex]
Therefore, both Selena and Drake, after correctly simplifying and substituting, arrived at the same result: [tex]\(\frac{1}{2}\)[/tex].
Conclusion:
Both Selena and Drake are correct in their evaluation of the expression [tex]\(\left(\frac{r s^{-2}}{r^2 s^{-3}}\right)^{-1}\)[/tex] when [tex]\(r = -1\)[/tex] and [tex]\(s = -2\)[/tex]. The final result for both of their methods is [tex]\(\frac{1}{2}\)[/tex].
