First, use this rule of exponents to rewrite the expression and eliminate some of the variables:
#a^color(red)(0) = 1#
#(pq^2r^-1s^6 * p^-3qr^2s^color(red)(0))/(p^5q^8r^color(red)(0)s^color(red)(0)) =>#
#(pq^2r^-1s^6 * p^-3qr^2 * 1)/(p^5q^8 * 1 * 1) =>#
#(pq^2r^-1s^6 * p^-3qr^2)/(p^5q^8)#
Next, rewrite the numerator as:
#((p * p^-3)(q^2 * q)(r^-1 * r^2)s^6)/(p^5q^8)#
Then, use these rules of exponents to simplify the numerator:
#a = a^color(red)(1)# and #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))# and #a^color(red)(1) = a#
#((p^color(red)(1) * p^color(blue)(-3))(q^color(red)(2) * q^color(blue)(1))(r^color(red)(-1) * r^color(blue)(2))s^6)/(p^5q^8) =>#
#(p^(color(red)(1)+color(blue)(-3))q^(color(red)(2)+color(blue)(1))r^(color(red)(-1)+color(blue)(2))s^6)/(p^5q^8) =>#
#(p^-2q^3r^color(red)(1)s^6)/(p^5q^8) =>#
#(p^-2q^3rs^6)/(p^5q^8)#
Next, rewrite the expression again as:
#(p^-2/p^5)(q^3/q^8)rs^6#
Now, use this rule of exponents to complete the simplification:
#x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))#
#(p^color(red)(-2)/p^color(blue)(5))(q^color(red)(3)/q^color(blue)(8))rs^6 =>#
#1/p^(color(blue)(5)-color(red)(-2))(1/q^(color(blue)(8)-color(red)(3)))rs^6 =>#
#(1/p^(color(blue)(5)+color(red)(2)))(1/q^(color(blue)(8)-color(red)(3)))rs^6 =>#
#1/p^7 * 1/q^5 * rs^6 =>#
#(rs^6)/(p^7q^5)#
