# How is multiplying and dividing rational expressions similar to multiplication and division of fractions. Give an example of each and compare the process please.

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### 1 Answer

The same rules apply in both cases: to multiply fractions or rational expressions the numerator of the answer is the product of the numerators and the denominator of the answer is the product of the denominators. To divide you multiply by the reciprocal.

Ex: `15/22 * 33/50` You could of course multiply 15 and 33, then 22 and 50 and simplify the result. `15/22 * 33/50 = 495/1100=9/20`

In practice it is easier to factor the numerators and denominators and "cancel" common factors.

`15/22 * 33/50 = (3*5)/(2*11) * (3*11)/(5*5*2)`

`=(3*3*5*11)/(2*2*5*5*11)=(3*3)/(2*2*5)=9/20` Note that we really noticed that `5/5,11/11` were just 1, and multiplying by 1 doesn't affect the product.

Ex: `(x^2+3x+2)/(x^2+7x+12)*(x^2+8x+15)/(x^2+5x+4)`

You could multiply the numerators and denominators and end up with a quartic over a quartic, and then hope to simplify the result. The easiest way is to factor first:

`((x+1)(x+2))/((x+3)(x+4))*((x+3)(x+5))/((x+1)(x+4))` and rewrite so that common factors are over each other:

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

`=((x+2)(x+5))/((x+4)(x+4))"or" (x^2+7x+10)/(x^2+8x+16)` though you usually leave in factored form.