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How Do You Multiply Quotients of Monomials by Canceling Factors?

Simplify this expression by canceling common factors:
[(2x2y)/(3yz)]•[(9y2z)/(16xd2)]

Summary

  1. Cancel out any common factors that you can before multiplying
  2. x2 is the same as x•x, so one of its x's cancels with the x in the denominator
  3. 9 is the same as 3•3, so one of its 3's cancels with the 3 in the denominator
  4. We can rewrite the monomials in the numerator and denominator based on what factors we've canceled out
  5. Everything in in the first monomial in the denominator canceled, so we can just write it as 1

Notes

    1. Dividing our fractions can allow us to multiply two simpler fractions than what we started with
    1. Factors of a monomial are all the variables or prime numbers that make up the monomial
    2. For example, the factors of 2x2y are 2, x2 and y
    1. To avoid problems, go through one by one and find the numbers and variables that cancel
    1. 2 is the first factor of 2x2y
    2. 16 is the first factor of 16xd2 on the bottom
    3. The 2 completely cancels out, and the 16 is reduced to 8 in the denominator
    1. x2 is a factor of 2x2y
    1. x2 is a factor of 2x2y
    2. x is the second factor in 16xd2
    3. x2 is just x•x, and one of those x's can cancel with the x in the denominator
    1. y is a factor of 2x2y
    2. There is a y in 2x2y and 3yz, so the two y's can cancel out completely
    1. 9 is a factor of 9y2z
    2. One of the 3's in 9 cancels with the 3 in 3yz
    1. y2 is a factor of 9y2z
    2. We can't use the y in 3yz to cancel with a y in y2 since we've already used the y in 3yz
    3. This is partly why we suggest going slowly, you don't want to reuse variables
    1. z is a factor of both 9y2z and 3yz, so it cancels out completely!
    1. We've checked all of the factors in the numerator with all the factors in the denominator
    1. Let's make sure we know what we've canceled by rewriting what is left over
    1. We canceled out a 2, x, y, 3, and z from the numerator
    2. That leaves us with an x, 3, y2 on the top
    3. We canceled out a 3, y, z, 2, and x from the denominator
    4. That leaves us with 8 and d2 on the bottom
    1. The first monomial in the numerator was reduced to just the x
    1. The first monomial in the denominator canceled completely, which is why we're putting a 1 there
    1. The second monomial in the numerator reduced to 3y2
    1. The second monomial in the denominator reduced to 8d2
    1. We can now multiply the two monomials in the numerator, and the monomial in the denominator
    1. Just multiply everything straight across and combine it into one term
    1. To avoid problems, go through one by one and find the numbers and variables that cancel
    2. After you cancel, rewrite the left over factors so that you can easily multiply them together