11.2 - Multiplication and division of algebraic fractions


Multiplication of algebraic fractions

The procedure for multiplying algebraic fractions is the same as the procedure for multiplying common fractions.


Multiplying two algebraic fractions produces a new algebraic fraction. Multiply the two numerators to get the new numerator and multiply the two denominators to get the new denominator:
Then simplify by reducing the new fraction to lowest terms.





Examples:




Division of algebraic fractions

The procedure for dividing algebraic fractions is the same as the procedure for dividing common fractions.


Replace the division by a fraction by the multiplication by the reciprocal of that fraction, like this:
Then carry out the multiplication of the two fractions as described above.


Notice that you take the reciprocal of the fraction on the bottom!


Here is why this procedure works:
The key is that instead of seeing a fraction divided by a fraction, look for a single fraction whose numerator and denominator just happen to be fractions. In the first step we multiplied this fraction by a UFOO whose numerator and denominator just happen to be fractions. The UFOO was chosen so the fractions in the denominator would cancel and give 1. After another simplification that left only the final multiplication of fractions.



Examples: Look for these three steps: (1) invert the bottom fraction, (2) multiply the fractions, (3) simplify.





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