First, rewrite this expression as:
#((-3 * 3)(p^-3 * p^3)(q * q^-4))^2 = (-9(p^-3 * p^3)(q * q^-4))^2#
Next, use these rules for exponents to simplify the terms within the outer parenthesis:
#a = a^color(red)(1)# and #x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) + color(blue)(b))#
#(-9(p^-3 * p^3)(q * q^-4))^2 = (-9(p^color(red)(-3) * p^color(blue)(3))(q^color(red)(1) * q^color(blue)(-4)))^2 =#
#(-9(p^(color(red)(-3)+color(blue)(3)))(q^(color(red)(1)+color(blue)(-4))))^2 = (-9(p^0)(q^-3))^2#
We can simplify the #p# terms using this rule of exponents:
#a^color(red)(0) = 1#
#(-9(p^color(red)(0))(q^-3))^2 = (-9(1)(q^-3))^2 = (-9q^-3)^2#
We can now use these rules for exponents to eliminate the outer exponent:
#a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#
#(-9q^-3)^2 = (-9^color(red)(1)q^color(red)(-3))^color(blue)(2) = -9^(color(red)(1)xxcolor(blue)(2))q^(color(red)(-3)xxcolor(blue)(2)) =#
#-9^2q^-6 = 81q^-6#
If we want the solution to have no negative exponents we can use this rule of exponents to further simplify the expression:
#x^color(red)(a) = 1/x^color(red)(-a)#
#81q^color(red)(-6) = 81/q^color(red)(- -6) = 81/q^color(red)(6)#
