For the first simplification we can use the rule for exponents:
#color(red)(a^0 = 1)#
#((-2x^3color(red)(y^0)z)/(4xz^2))^5 -> ((-2x^3color(red)(1)z)/(4xz^2))^5 -> ((-2x^3z)/(4xz^2))^5#
Next simplification is to reduce the constants:
#((-2x^3z)/(4xz^2))^5 -> ((-2x^3z)/((color(red)(2*2))xz^2))^5 ->#
#((-color(red)(cancel(color(black)(2)))x^3z)/((color(red)(cancel(2)*2))xz^2))^5 -> ((-x^3z)/(2xz^2))^5#
Next, we can take advantage of two other rules for exponents:
-
#color(blue)(x^a/x^b = x^(a-b)#
-
#color(red)(x^a/x^b = 1/x^(b-a)#
#((-color(blue)(x^3)color(red)(z))/(2color(blue)(x)color(red)(z^2)))^5 -> ((-color(blue)(x^(3-1)))/(2color(red)(z^(2-1))))^5 -> ((-x^2)/(2z^1))^5#
Now, we can use yet another rule of exponents to further simplify this expression:
#color(red)((x^a)^b) = x^(a*b)#
#((-x^2)/(2z^1))^5 -> (-x^(color(red)(2*5)))/(2^color(red)(5)z^color(red)(1*5)) -> (-x^10)/(32z^5)#