Simplifying expressions

Question

How would I, step by step, simplify an expression that looks like the following:

 

[15a^8b^6]      [2b^2]

 _________       ______

   [ab^3]        [3a^3b^8]

 

(assume that all variable in the denominator are nonzero)

Answer 1

You want to use the rules of exponents.

 

1. (x^a) (x^b) = x^(a+b)

 e.g. (x^2) (x^3) = x^5

 Why: (x^2) (x^3) = (x x) (x x x) = (x x x x x) = x^5

 

2. (x^a) / (x^b) = x^(a-b)

 e.g. (x^5) / (x^3) = (x^2)

 Why: (x^5) / (x^3) = (x x x x x) / (x x x) = (x x) [(x x x) / (x x x)] = (x x) (1) = x^2

 

3. (x^a)^b = x^(ab)

 e.g. (x^3)^2 = x^6

 Why: (x^3)^2 = (x^3) (x^3) = x^6

 

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In your problem, group all the coefficients (real numbers), and like variables together on the top and bottom:

[(15) (2)   (a^8)   (b^6) (b^2)]   /   [(3)   (a) (a^3)   (b^3) (b^8)]

 

Then apply rule 1 followed by rule 2.

   [30 a^8 b^8] / [3 a^4 b^11]

= (10 a^4) / (b^3)   <-- here b^3 is left in the denominator because b^8 cancels on the top and bottom, leaving b^3 on the bottom, and none (i.e. b^0) on the top.