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How Do You Simplify a Mixed Expression Over a Mixed Expression?

Simplify ((1+(4/(x+3))/(1+(2/(x+5))

Summary

  1. First, find the LCD of the main numerator and denominator
  2. Realize that '1' just has a denominator of '1'
  3. Multiply the numerators together, then multiply the denominators by each other
  4. Cancelling out common factors will help simplify our complex fraction
  5. Multiplying and distributing our polynomials will help us to simplify further
  6. Rewriting polynomials as factors can make cancelling out common factors a breeze!

Notes

    1. The numerator and denominator of the complex fraction each have their own LCD
    1. The numerator of the complex fraction is: (1+(4/(x+3))
    2. Remember, '1' is the same as '1/1', so its denominator is '1'
    3. For the fraction '4/(x+3)', the numerator is '4' and the denominator is 'x+3'
    1. We found two denominators, '1' and 'x+3'
    2. Between these two denominators, the LCD is 'x+3'
    1. The denominator of the complex fraction is: (1+(2/(x+5))
    2. Remember, '1' is the same as '1/1', so its denominator is '1'
    3. For the fraction '2/(x+5)', the numerator is '2' and the denominator is 'x+5'
    1. We found two denominators, '1' and 'x+5'
    2. Between these two denominators, the LCD is 'x+5'
    1. Our two LCDs are 'x+3' and 'x+5'
    1. Our two LCDs are 'x+3' and 'x+5'
    2. '(x+3)(x+5)/(x+3)(x+5)' is a fancy form of '1'
    1. The numerator of the complex fraction is: (1+(4/(x+3))
    2. Our two LCDs are 'x+3' and 'x+5'
    3. We're multiplying the complex fraction by '(x+3)(x+5)/(x+3)(x+5)', so we multiply the numerators together first
    1. The denominator of the complex fraction is: (1+(2/(x+5))
    2. Our two LCDs are 'x+3' and 'x+5'
    3. We're multiplying the complex fraction by '(x+3)(x+5)/(x+3)(x+5)', so now we multiply the denominators together
    1. So we want to find some way to simplify it from this point
    1. Up to this point, our main numerator looks like:
    2. (x+3)(x+5)+(4(x+3)(x+5))/(x+3)
    3. '(4(x+3)(x+5))/(x+3)' can simplify to '4(x+5)' since the '(x+3)' terms cancel out
    1. Up to this point, our main denominator looks like:
    2. (x+3)(x+5)+(2(x+3)(x+5))/(x+5)
    3. '(2(x+3)(x+5))/(x+5)' can simplify to '2(x+3)' since the '(x+5)' terms cancel out
    1. This is equivalent to our original complex fraction, it's just easier to work with
    1. Up to this point, our main numerator looked like:
    2. (x+3)(x+5)+4(x+5)
    1. Up to this point, our main denominator looked like:
    2. (x+3)(x+5)+2(x+3)
    1. Our original complex fraction now looks like:
    2. (x2+8x+15+4x+20)/
    3. (x2+8x+15+2x+6)
    1. Our original complex fraction now looks like:
    2. (x2+8x+15+4x+20)/
    3. (x2+8x+15+2x+6)
    1. Up to this point, our original complex fraction looked like:
    2. (x2+8x+15+4x+20)/
    3. (x2+8x+15+2x+6)
    4. '8x' and '4x' are like terms, so we add them together
    5. '15' and '20' can also be added together
    6. In the denominator, '8x' and '2x' are like terms, so we combine them
    7. '15' and '6' can also be added together in the denominator
    1. Up to this point, our main numerator looked like:
    2. x2+12x+35
    3. If you multiply '(x+7)' by '(x+5)', you should get 'x2+12x+35'
    1. Up to this point, our main denominator looked like:
    2. x2+10x+21
    3. If you multiply '(x+7)' by '(x+3)', you should get 'x2+10x+21'
    1. Up to this point, our main fraction looked like:
    2. (x+7)(x+5)/(x+7)(x+3)
    3. Since there is one '(x+7)' term in both the numerator and denominator, they cancel out
    1. Our original complex fraction now looks like:
    2. (x+5)/(x+3)
    1. Our original complex fraction now looks like:
    2. (x+5)/(x+3)
    1. Our original complex fraction now looks like:
    2. (x+5)/(x+3)