To determine which expression is equivalent to the given expression [tex]\(\left(\frac{r^{-5}}{r^{-9}}\right)^{1 / 2}\)[/tex], let's go through the steps of simplifying it:
1. Step 1: Apply the division property of exponents
[tex]\[
\frac{r^{-5}}{r^{-9}} = r^{-5 - (-9)} = r^{-5 + 9} = r^4
\][/tex]
2. Step 2: Simplify the exponents within the parentheses
This simplifies our original expression to:
[tex]\[
\left(r^4\right)^{1 / 2}
\][/tex]
3. Step 3: Apply the power of a power property of exponents
When an exponent is raised to another exponent, we multiply the exponents:
[tex]\[
\left(r^4\right)^{1 / 2} = r^{4 \cdot \frac{1}{2}} = r^{4 / 2} = r^2
\][/tex]
Thus, the expression [tex]\(\left(\frac{r^{-5}}{r^{-9}}\right)^{1 / 2}\)[/tex] simplifies to [tex]\(r^2\)[/tex].
Therefore, the correct answer is:
C) [tex]\(r^2\)[/tex]
