First, use these rules of exponents to simplify the left term in the numerator:
#a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#
#((2a^2b)^2(3ab^3c))/(4a^4b^8c^2) => ((2^color(red)(1)a^color(red)(2)b^color(red)(1))^color(blue)(2)(3ab^3c))/(4a^4b^8c^2) =>#
#((2^(color(red)(1)xxcolor(blue)(2))a^(color(red)(2)xxcolor(blue)(2))b^(color(red)(1)xxcolor(blue)(2)))(3ab^3c))/(4a^4b^8c^2) =>#
#((2^2a^4b^2)(3ab^3c))/(4a^4b^8c^2) =>#
#((4a^4b^2)(3ab^3c))/(4a^4b^8c^2)#
Next, rewrite the expression as:
#(4 * 3)/4((a^4a)/a^4)((b^2b^3)/(b^8))(c/c^2) =>#
#(color(red)(cancel(color(black)(4))) * 3)/color(red)(cancel(color(black)(4)))((color(red)(cancel(color(black)(a^4)))a)/color(red)(cancel(color(black)(a^4))))((b^2b^3)/(b^8))(c/c^2) =>#
#3a((b^2b^3)/(b^8))(c/c^2)#
Then, use this rule of exponents to simplify the numerator of the #b# terms:
#x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#
#3a((b^2b^3)/(b^8))(c/c^2)#
#3a((b^color(red)(2)b^color(blue)(3))/(b^8))(c/c^2) => 3a((b^(color(red)(2)+color(blue)(3)))/(b^8))(c/c^2) =>#
#3a(b^5/(b^8))(c/c^2)#
Now, use these rules to simplify the #b# and #c# terms:
#a = a^color(red)(1)# and #x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))# and #a^color(red)(1) = a#
#3a(b^5/(b^8))(c/c^2) => 3a(b^color(red)(5)/(b^color(blue)(8)))(c^color(red)(1)/c^color(blue)(2)) =>#
#3a(1/(b^(color(blue)(8)-color(red)(5))))(1/c^(color(blue)(2)-color(red)(1))) => 3a(1/b^3)(1/c^1) => 3a(1/b^3)(1/c) =>#
#(3a)/(b^3c)#
