SOLUTION: Simplify the rational expression. {{{((1)/(x)+(3)/(x^2))/(x+(27)/(x^2))}}} Can you help me.

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Question 425600: Simplify the rational expression.



Can you help me.
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

As often happens in Math, there is more than one way to simplify an expression like this. The way I prefer is:
  1. Find the Lowest Common Denominator (LCD) of all the "little" fractions.
  2. Multiply the numerator and denominator of the "big" fraction by this LCD.
  3. Simplify.

Your "little" denominators are x, and . The LCD of these three is . So we'll multiply the numerator of the big fraction by :

In both the numerator and denominator we will use the Distributive Property to multiply:

Every denominator of a "little" fraction cancels out! leaving us with:

Now we simplify. The numerator and denominator are simplified so allt here is to do is to try to reduce the fraction. As always reducing fractions involves canceling factors that are common to the numerator and denominator. We need to know what the factors are to see if there are any that will cancel. So we start reducing by factoring. The numerator will not factor (except factoring out a 1). The denominator is a sum of cubes, , so we can use the to factor it:

And we can now that there is a factor that will cancel:

leaving:


An alternate way to do this is to
  1. Add the terms in the "big" numerator and denomiantor (getting common denominators first, of course.
  2. Step 1 turns the numerator and denominator of the "big" fraction into fractions. Since a fraction is another way of expressing a division, we now have the numerator fraction divided by the denominator fraction. As you learned long ago, dividing fractions is done by multiplying by the reciprocal. So the next step is to change the expression to multiplying by the reciprocal. Here's a simplified example:
  3. Try to cancel factors before you multiply.
  4. Multiply
  5. Reduce the fraction if possible.
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