how do you simplify ((((12^(x-1))*(13^(y+1)))^2)-(((12^x)*(13^y))^2))^(1/2)

Answer 1

problem: ((((12^(x-1))*(13^(y+1)))^2)-(((12^x)*(13^y))^2))^(1/2)

visualize layers: [ ( [ (12^(x-1))*(13^(y+1)) ]^2 ) - ( [ (12^x)*(13^y) ]^2 ) ]^(1/2), this is just some (A-B)^(1/2), so we will break down A and B seperate

A: ( [ (12^(x-1))*(13^(y+1)) ]^2 )

12^(x-1) = 12^x * 12^(-1) = 12^x * (1/12)

13^(y+1) = 13^y * 13^1 = 13^y * 13

(12^(x-1))*(13^(y+1)) = (12^x * (1/12)) * (13^y * 13) = 12^x * (1/12) * 13^y * 13

[ (12^(x-1))*(13^(y+1)) ]^2 = (12^x * (1/12) * 13^y * 13)^2 = 12^(2x) * (1/12)^2 * 13^(2y) * 13^2 = 12^(2x) * 1/144 * 13^(2y) * 169 = (169/144) * 12^(2x) * 13^(2y)

B: ( [ (12^x)*(13^y) ]^2 ) = 12^(2x) * 13^(2y)

A-B: (169/144) * 12^(2x) * 13^(2y) - 12^(2x) * 13^(2y) = (25/144) * 12^(2x) * 13^(2y)

finally: (A-B)^(1/2) = ((25/144) * 12^(2x) * 13^(2y))^(1/2) = (25/144)^(1/2) * 12^(2x*1/2) * 13^(2y*1/2) =

(5/12) * 12^x * 13^y = 5 * 12^(x-1) * 13^y

Hence, ((((12^(x-1))*(13^(y+1)))^2)-(((12^x)*(13^y))^2))^(1/2) = 5 * 12^(x-1) * 13^y