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How Do You Reduce Common Factors in a Rational Expression?

Reduce the common factors in this rational expression:
[2x2y)/(3yz)]•[(9y2z)/(16xd2)]

Summary

  1. We can use the Associative Property of Multiplication to regroup terms over one another
  2. Terms with the same variable should be grouped together
  3. Any number or variable divided by itself is equal to 1, so let's change those fractions to 1!
  4. The Identity Property of Multiplication allows us to get rid of the 1's
  5. Let's multiply together the fractions that are left!

Notes

    1. We can use the Associative Property on the terms in the numerator and then on the terms in the denominator
    1. It'll be easier to figure out how to do the multiplication if we regroup terms with common factors
    2. For example, we can put anything with an x in the numerator over anything with an x in the denominator
    1. Right now we're grouping just the numbers together using the Associative Property of Multiplication
    1. Now we're grouping terms with the same variables together using the Associative Property of Multiplication
    1. Now we're grouping terms with the same variables together using the Associative Property of Multiplication
    1. Now we're grouping terms with the same variables together using the Associative Property of Multiplication
    1. Now we're grouping terms with the same variables together using the Associative Property of Multiplication
    1. We want to factor each numerator and denominator
    2. This will allow us to see exactly how each factor cancels
    1. 2•9 is the same as 2•3•3 when we factor the 9
    2. 3•16 is the same as 3•2•8 when we factor the 16
    3. 2 and 3 are factors in both the numerator and denominator
    4. Use the Associative Property of Multiplication to rearrange the bottom so that the 2's and 3's are under each other
    1. We can separate y•y2 and allow the first y to be paired with a y in the denominator
    1. There are no 'd' variables in the numerator, so there's no way to separate the d2 in such a way that we can cancel
    1. These are some fractions that can completely cancel out!
    1. Anything divided by itself is just 1, so let's change what fractions we can into just 1
    1. This includes 2/2, 3/3, x/x, y/y, and z/z
    1. Since anything divided by itself is just 1, we can rewrite 2/2, 3/3, x/x, y/y, and z/z as just 1
    1. We can take the fractions that are left over, and just combine them into one big fraction
    1. Anything divided by itself is just 1, so we changed what fractions we could into just 1
    2. The Identity Property of Multiplication says that anything times 1 is itself, so we're left with 3/8
    1. Anything divided by itself is just 1, so we changed what fractions we could into just 1
    2. The Identity Property of Multiplication says that anything times 1 is itself, so we're left with x/1 and y/1
    1. z over z becomes 1, and 1 times anything is itself
    1. The numerators are 3, x, y2 and 1
    2. 3•x•y2•1=3xy2
    1. The denominators are 8, 1, 1, and d2
    2. 8•1•1•d2=8d2
    1. Common factors are just factors that occur in both the numerator and denominator of a fraction