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The product of two fractions is defined as follows.The product of two fractions is a fraction whose numerator is the product of the numerators and whose denominator is the product of the denominators of the given fractions.Example 2 Find the LCD of the fractions Solution Following the method of Example 1, we get Thus, the LCD is x Example 3 Write the sums of and as single terms. We build each fraction to a fraction with 10 as the denominator. We build each fraction to a fraction with denominator (x 2)(x - 1), inserting parentheses as needed, and get Now that we have like denominators, we can add the numerators, simplify, and obtain Example 5 Write the sum of as a single term.
In symbols, Example 1 As in multiplication, when fractions in a quotient have signs attached, it is advisable to proceed with the problem as if all the factors were positive and then attach the appropriate sign to the solution.
Example 2 Some quotients occur so frequently that it is helpful to recognize equivalent forms directly.
We now build each fraction to fractions with this denominator and get We can now add the numerators, simplify, and obtain Common Errors Note that we can only add fractions with like denominators.
Thus, Also, we only add the numerators of fractions with like denominators.
To find the LCD: Example 1 Find the lowest common denominator of the fractions Solution The lowest common denominator for contains among its factors the factors of 12, 10, and 6. (This number is the smallest natural number that is divisible by 12, 10, and 6.) The LCD of a set of algebraic fractions is the simplest algebraic expression that is a multiple of each of the denominators in the set.
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Thus, the LCD of the fractions because this is the simplest expression that is a multiple of each of the denominators.
Example 2 Find the product of Solution First, we divide the numerator and denominator by the common factors to get Now, multiplying the remaining factors of the numerators and denominators yields If a negative sign is attached to any of the factors, it is advisable to proceed as if all the factors were positive and then attach the appropriate sign to the result.
A positive sign is attached if there are no negative signs or an even number of negative signs on the factors; a negative sign is attached if there is an odd number of negative signs on the factors.
In algebra, we often rewrite an expression such as as an equivalent expression .
Use whichever form is most convenient for a particular problem.