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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 symbols, Any common factor occurring in both a numerator and a denominator of either fraction can be divided out either before or after multiplying.
Example 1 Find the product of Solution The same procedures apply to fractions containing variables.
In algebra, we often rewrite an expression such as as an equivalent expression .
Use whichever form is most convenient for a particular problem.
This is precisely the same notion as that of dividing one integer by another; a ÷ b is a number q, the quotient, such that bq = a. To solve this equation for q, we multiply each member of the equation by .
Thus, In the above example, we call the number the reciprocal of the number .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, Example 1 When subtraction is involved, it is helpful to change to standard form before adding.Example 2 We must be especially careful with binomial numerators.For example, we should rewrite where the entire numerator is enclosed within parentheses.In Section 6.3, we added fractions with like denominators.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.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.