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Multiplying and Dividing Fractions

Examples with Solutions

Example 1: Multiply

solution: This is just another way of posing the problem: Simplify:

Proceeding as in the previous example, we get

as the final solution. We could have done this last step as

to get the same final result, once factors are repositioned between the numerator and denominator to get rid of negative exponents.

Example 2: Simplify .

solution: This is very similar to the expressions handled in the first two examples. Proceeding in the same fashion, we get

(This is what the actual multiplication of the two fractions amounts to. Now this result must be simplified.)

(Here we expand the numerical factors into products of prime factors, and we also sort out the various literal factors.)

(Cancel the common numerical factors and combine powers of the literal symbols.)

(This is a fully simplified form, but it contains negative exponents.)

as the final result with negative exponents eliminated.

Example 3: Simplify .

solution: This is one fraction divided by another. Following the pattern given at the beginning of this document, we know that the first step here is to rewrite the expression as the first fraction multiplied by the reciprocal of the second fraction:

Now the remainder of the work is to simplify this multiplication, exactly as we dealt with the first three examples. So

as the final answer.

A very common error here is to start by correctly rewriting the original division problem as a multiplication, but then doing the multiply step in a totally bizarre way – numerators with denominators, as in

But this is totally wrong! When we multiply two fractions, it is always numerator times numerator and denominator times denominator, regardless of where the multiplication problem originally came from. If you examine the eventual result that would be obtained here, you’ll see that it amounts to what we would get if we had multiplied the two original fractions together, rather than dividing the first fraction by the second one. In other words, what has effectively been done by using this erroneous method is to change the original division symbol to a multiplication symbol. This must be an error. So, always remember: dividing by a fraction is equivalent to multiplying by its reciprocal, and multiplying is always done the same way, regardless of from where the original fractions were obtained.

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