Evaluate each expression for \( r=-3 \) and \( s=5 \). \( \begin{array}{llll}\text { 37. } r^{-3} & \text { 38. } s^{-3} & \text { 39. } \frac{3 r}{s^{-2}} & \text { 40. } \frac{s^{0}}{r^{-2}} \\ \begin{array}{lll}\text { 41. } 4 s^{-1} & \text { 42. } r^{0} s^{-2} & \text { 43. } r^{-4} s^{2}\end{array}\end{array} \begin{array}{l}\text { 44. } 2^{-4} r^{3} s^{-2}\end{array} \)

Let's evaluate each expression step by step for \( r = -3 \) and \( s = 5 \). 1. **\( r^{-3} \)**: \( (-3)^{-3} = \frac{1}{(-3)^3} = \frac{1}{-27} = -\frac{1}{27} \). 2. **\( s^{-3} \)**: \( 5^{-3} = \frac{1}{5^3} = \frac{1}{125} \). 3. **\( \frac{3 r}{s^{-2}} \)**: First, compute \( s^{-2} \): \( 5^{-2} = \frac{1}{25} \). Therefore, \( \frac{3 (-3)}{\frac{1}{25}} = 3(-3) \cdot 25 = -9 \cdot 25 = -225 \). 4. **\( \frac{s^{0}}{r^{-2}} \)**: Note that \( s^{0} = 1 \) and \( r^{-2} = \frac{1}{(-3)^2} = \frac{1}{9} \). Thus, \( \frac{1}{\frac{1}{9}} = 9 \). 5. **\( 4 s^{-1} \)**: \( s^{-1} = \frac{1}{5} \). Therefore, \( 4 \cdot \frac{1}{5} = \frac{4}{5} \). 6. **\( r^{0} s^{-2} \)**: \( r^{0} = 1 \) and \( s^{-2} = \frac{1}{25} \). Hence, \( 1 \cdot \frac{1}{25} = \frac{1}{25} \). 7. **\( r^{-4} s^{2} \)**: \( r^{-4} = \frac{1}{(-3)^4} = \frac{1}{81} \) and \( s^{2} = 5^2 = 25 \). Therefore, \( \frac{1}{81} \cdot 25 = \frac{25}{81} \). 8. **\( 2^{-4} r^{3} s^{-2} \)**: \( 2^{-4} = \frac{1}{16} \), \( r^{3} = (-3)^3 = -27 \), and \( s^{-2} = \frac{1}{25} \). Combining these gives \( \frac{1}{16} \cdot (-27) \cdot \frac{1}{25} = -\frac{27}{400} \). Here are all the evaluated expressions summarized: - **37.** \( r^{-3} = -\frac{1}{27} \) - **38.** \( s^{-3} = \frac{1}{125} \) - **39.** \( \frac{3 r}{s^{-2}} = -225 \) - **40.** \( \frac{s^{0}}{r^{-2}} = 9 \) - **41.** \( 4 s^{-1} = \frac{4}{5} \) - **42.** \( r^{0} s^{-2} = \frac{1}{25} \) - **43.** \( r^{-4} s^{2} = \frac{25}{81} \) - **44.** \( 2^{-4} r^{3} s^{-2} = -\frac{27}{400} \)