For example, we should rewrite where the entire numerator is enclosed within parentheses.
In Section 6.3, we added fractions with like denominators.
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.
Similarly, because 3 is not a factor of the entire numerator 3y 2.
In dividing one fraction by another, we look for a number that, when multiplied by the divisor, yields the dividend.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.Thus, are equivalent to from which we obtain Sometimes, the fractions have denominators that are binomials. Solution First we factor the denominators in order to obtain the LCD.In this section, we will add fractions with unlike denominators.LEAST COMMON DENOMINATOR In general, the smallest natural number that is a multiple of each of the denominators of a set of fractions is called the lowest common denominator (LCD) of the set of fractions. If the LCD is not immediately evident, we can use a special procedure to find it.Example 3 When the fractions contain algebraic expressions, it is necessary to factor wherever possible and divide out common factors before multiplying. Solution First, we must factor the numerators and denominators to get Now, dividing out common factors yields We now multiply the remaining factors of the numerators and denominators to obtain Note that when writing fractional answers, we will multiply out the numerator and leave the denominator in factored form.Very often, fractions are more useful in this form.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.In general, the reciprocal of a fraction is the fraction .That is, we obtain the reciprocal of a fraction by "inverting" the fraction.