First, rewrite this expression as:
#((2/-4)(a^-2/a^-2)(b^4/b^-5)(c^2/c^-7))^-1 =>#
#(-1/2 * 1(b^4/b^-5)(c^2/c^-7))^-1 =>#
#(-1/2(b^4/b^-5)(c^2/c^-7))^-1#
Next, use this rule of exponents to simplify the terms within the parenthesis:
#x^color(red)(a)/x^color(blue)(b) = 1/x^(color(blue)(b)-color(red)(a))#
#(-1/2(b^4/b^-5)(c^2/c^-7))^-1 => (-1/2(b^color(red)(4)/b^color(blue)(-5))(c^color(red)(2)/c^color(blue)(-7)))^-1 =>#
#(-1/2(1/b^(color(blue)(-5)-color(red)(4)))(1/c^(color(blue)(-7)-color(red)(2))))^-1 => (-1/(2b^-9c^-9))^-1#
Now, use these rules of exponents to complete the simplification:
#a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))# and #1/x^color(red)(a) = x^color(red)(-a)#
#(-1/(2b^-9c^-9))^-1 => (-1/(2^color(red)(1)b^color(red)(-9)c^color(red)(-9)))^color(blue)(-1) => -1/(2^(color(red)(1) xx color(blue)(-1))b^(color(red)(-9) xx color(blue)(-1))c^(color(red)(-9) xx color(blue)(-1))) =>#
#-1/(2^color(red)(-1)b^9c^9) => -2^color(red)(- -1)/(b^9c^9) =>#
#-2/(b^9c^9)#
