Express the following in simplified form without zero and negative exponents (2a^3 b^-3 c^-1 / 3a^-2 b^-2 c^3)^3Show complete solution to explain the answer.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to simplify: (2a^3 b^-3 c^-1 / 3a^-2 b^-2 c^3)^3

(2a^3 b^-3 c^-1 / 3a^-2 b^-2 c^3)^3

=> (2a^(3+2)*b^(-3+2)*c^(-1-3)/3)^3

=> (2a^5*b^-1*c^-4/3)^3

=> (8/27)*a^15*b^-3*c^-12

=> (8/27)*a^15/(b^3*c^12)

The required simplified form is (8/27)*a^15/(b^3*c^12)

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samhouston | Middle School Teacher | (Level 1) Associate Educator

Posted on

According to exponential law, when powers are divided, their exponents are subtracted.

n^a / n^b = n^a-b

Take a look at each variable by itself.  Subtract the exponents.  Remember that minus negative becomes addition.

a^3 / a^-2 = a^(3 - -2) = a^5

b^-3 / b^-2 = b^(-3 - -2) = b^-1

c^-1 / c^3 = c^(-1 - 3) = c^-4

The variables with positive exponents will be part of the numerator and the variables with negative exponents will be part of the denominator.

numerator:  2a^5

denominator:  3b^1c^4

The entire fraction, both numerator and denominator, is raised to the 3rd power.

numerator: (2a^5)^3

denominator: (3b^1c^4)^3

According to exponential law, to find the power of a power, you multiply the exponents.  Remember to also distribute the exponent 3 to the contants.

numerator:  2^3 * a^(5*3) = 8a^15

denominator:  3^3 * b^(1*3) * c^(4*3) = 27b^3c^12

The final simplified answer is...

(8a^15) / (27b^3c^12)

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