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i need help

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Aug 14, 2020

#1
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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)

Aug 14, 2020