Simplify each expression, assuming that no variable equals zero. Write your answer with positive exponents only. \( \begin{array}{llll}\text { 39. } y^{5} y^{2} & \text { 40. }-2 z^{3} z^{5} & \text { 41. }-2 y^{3}\left(5 x y^{4}\right) & \text { 42. } 6 x^{5} \cdot 3 x^{5} \cdot x^{0} \\ \text { 43. } \frac{m^{9}}{m^{5}} & \text { 44. } \frac{b b^{4}}{b^{2}} & \text { 45. } \frac{x^{2} x^{-5}}{x^{4}} & \text { 46. } \frac{s^{5} t^{2}}{s t^{-4}} \\ \text { 47. }\left(2 x^{4} y\right)^{3} & \text { 48. }\left(3 s t^{12}\right)^{3} & \text { 49. }\left(-5 w^{4} v^{5}\right)^{2} & \text { 50. }\left(-3 x^{2} y^{7}\right)^{3} \\ \text { 51. }\left(\frac{-2 z^{2}}{x^{3}}\right)^{7} & \text { 52. }\left(\frac{2 b^{4}}{-a^{2}}\right)^{3} & \text { 53. }\left(\frac{-2 p^{5} q^{-4}}{q^{3}}\right)^{3} & \text { 54. }\left(\frac{3 m^{2} n^{3}}{m^{-1}}\right)^{5}\end{array} \)

Let's simplify each expression step by step: **39.** \( y^{5} y^{2} \) Using the property \( a^m a^n = a^{m+n} \): \[ y^{5} y^{2} = y^{5+2} = y^{7} \] **40.** \( -2 z^{3} z^{5} \) Using the same property: \[ -2 z^{3} z^{5} = -2 z^{3+5} = -2 z^{8} \] **41.** \( -2 y^{3}(5 x y^{4}) \) Distributing: \[ -2 y^{3} \cdot 5 x y^{4} = -10 x y^{3+4} = -10 x y^{7} \] **42.** \( 6 x^{5} \cdot 3 x^{5} \cdot x^{0} \) First, simplify the coefficients and then the exponents: \[ (6 \cdot 3) x^{5+5+0} = 18 x^{10} \] **43.** \( \frac{m^{9}}{m^{5}} \) Using the property \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{m^{9}}{m^{5}} = m^{9-5} = m^{4} \] **44.** \( \frac{b b^{4}}{b^{2}} \) Using the same property: \[ \frac{b^{1} b^{4}}{b^{2}} = \frac{b^{1+4}}{b^{2}} = \frac{b^{5}}{b^{2}} = b^{5-2} = b^{3} \] **45.** \( \frac{x^{2} x^{-5}}{x^{4}} \) First, simplify the numerator: \[ \frac{x^{2} x^{-5}}{x^{4}} = \frac{x^{2-5}}{x^{4}} = \frac{x^{-3}}{x^{4}} = x^{-3-4} = x^{-7} \] Since we want positive exponents: \[ x^{-7} = \frac{1}{x^{7}} \] **46.** \( \frac{s^{5} t^{2}}{s t^{-4}} \) Simplifying: \[ \frac{s^{5} t^{2}}{s^{1} t^{-4}} = \frac{s^{5-1} t^{2-(-4)}}{1} = s^{4} t^{2+4} = s^{4} t^{6} \] **47.** \( \left(2 x^{4} y\right)^{3} \) Using the power of a product property: \[ 2^{3} (x^{4})^{3} (y)^{3} = 8 x^{12} y^{3} \] **48.** \( \left(3 s t^{12}\right)^{3} \) Using the same property: \[ 3^{3} (s)^{3} (t^{12})^{3} = 27 s^{3} t^{36} \] **49.** \( \left(-5 w^{4} v^{5}\right)^{2} \) Using the same property: \[ (-5)^{2} (w^{4})^{2} (v^{5})^{2} = 25 w^{8} v^{10} \] **50.** \( \left(-3 x^{2} y^{7}\right)^{3} \) Using the same property: \[ (-3)^{3} (x^{2})^{3} (y^{7})^{3} = -27 x^{6} y^{21} \] **51.** \( \left(\frac{-2 z^{2}}{x^{3}}\right)^{7} \) Using the property of powers: \[ \frac{(-2)^{7} (z^{2})^{7}}{(x^{3})^{7}} = \frac{-128 z^{14}}{x^{21}} = -128 z^{14} x^{-21} \] **52.** \( \left(\frac{2 b^{4}}{-a^{2}}\right)^{3} \) Using the property of powers