Simplify the following:
((4/(a^6))^3 (a^4 b)^3)/(12 a^4 (a^3 b^2)^4)
Multiply each exponent in a^4 b by 3:
((4/(a^6))^3 a^(3×4) b^3)/(12 a^4 (a^3 b^2)^4)
3×4 = 12:
((4/(a^6))^3 a^12 b^3)/(12 a^4 (a^3 b^2)^4)
Multiply each exponent in a^3 b^2 by 4:
((4/(a^6))^3 a^12 b^3)/(12 a^4 a^(4×3) b^(4×2))
4×2 = 8:
((4/(a^6))^3 a^12 b^3)/(12 a^4 a^(4×3) b^8)
4×3 = 12:
((4/(a^6))^3 a^12 b^3)/(12 a^4 a^12 b^8)
Multiply each exponent in 4/(a^6) by 3:
(4^3 a^(-6×3) a^12 b^3)/(12 a^4 a^12 b^8)
3 (-6) = -18:
(4^3 a^(-18) a^12 b^3)/(12 a^4 a^12 b^8)
4^3 = 4×4^2:
(4×4^2 a^12 b^3)/(a^18×12 a^4 a^12 b^8)
4^2 = 16:
(4×16 a^12 b^3)/(a^18×12 a^4 a^12 b^8)
4×16 = 64:
(64 a^12 b^3)/(a^18×12 a^4 a^12 b^8)
(64 a^12 b^3)/(a^18×12 a^4 a^12 b^8) = a^12/a^12×(64 b^3)/(a^18×12 a^4 b^8) = (64 b^3)/(a^18×12 a^4 b^8):
(64 b^3)/(a^18×12 a^4 b^8)
The gcd of 64 and 12 is 4, so (64 b^3)/(a^18×12 a^4 b^8) = ((4×16) b^3)/(a^18 (4×3) a^4 b^8) = 4/4×(16 b^3)/(a^18×3 a^4 b^8) = (16 b^3)/(a^18×3 a^4 b^8):
(16 b^3)/(a^18×3 a^4 b^8)
Combine powers. (16 b^3)/(a^18×3 a^4 b^8) = (16 a^(-4 - 18) b^(3 - 8))/3:
(16 a^(-4 - 18) b^(3 - 8))/3
-4 - 18 = -22:
(16 a^(-22) b^(3 - 8))/3
3 - 8 = -5:
(16 b^(-5)) / (3 a^22)
