8. Radicals - Multiply and Divide Radicals Objective: Multiply and divide radicals using the product and quotient rules of radicals. Multiplying radicals is very simple if the index on all the radicals match. The prodcut rule of radicals which we have already been using can be generalized as follows: Product Rule of Radicals: a m b c m d = ac m bd Another way of stating this rule is we are allowed to multiply the factors outside the radical and we are allowed to multiply the factors inside the radicals, as long as the index matches. This is shown in the following example. Example 1. 1 6 0 8 0 1 0 1 0 1 Multiply outside and inside the radical Simplify the radical, divisible by Take the square root where possible The same process works with higher roots Example. 18 6 1 1 0 1 1 6 Multiply outside and inside the radical Simplify the radical, divisible by Take cube root where possible 1
When multiplying with radicals we can still use the distributive property or FOIL just as we could with variables. Example. 6 ( 1) Distribute, following rules for multiplying radicals 1 60 90 Simplify each radical, finding perfect square factors 1 1 9 Take square root where possible 1 1 1 Example. ( )( + 6 6) 0 + 6 0 8 0 1 18 +6 0 8 0 1 9 +6 0 8 0 1 0 + 6 0 8 0 6 16 0 FOIL, following rules for multiplying radicals Simplify radicals, find perfect square factors Take square root where possible Combine like terms World View Note: Clay tablets have been discovered revealing much about Babylonian mathematics dating back from 1800 to 1600 BC. In one of the tables there is an approximation of accurate to five decimal places (1.11) Example. ( 6)( 8 ) 1 16 1 1 1 16 1 1 16 1 1 16 FOIL, following rules for multiplying radicals Simplify radicals, find perfect square factors Take square root where possible Multiply coefficient As we are multiplying we always look at our final solution to check if all the radi-
cals are simplified and all like radicals or like terms have been combined. Division with radicals is very similar to multiplication, if we think about division as reducing fractions, we can reduce the coefficients outside the radicals and reduce the values inside the radicals to get our final solution. Quotient Rule of Radicals: a m b c m d = a c m b d Example 6. 1 8 0 Reduce 1 0 and 8 9 Simplify radical, is divisible by Take the cube root of by dividing by andrespectively There is one catch to dividing with radicals, it is considered bad practice to have a radical in the denominator of our final answer. If there is a radical in the denominator we will rationalize it, or clear out any radicals in the denominator. We do this by multiplying the numerator and denominator by the same thing. The problems we will consider here will all have a monomial in the denominator. The way we clear a monomial radical in the denominator is to focus on the index. The index tells us how many of each factor we will need to clear the radical. For example, if the index is, we will need of each factor to clear the radical. This is shown in the following examples. Example. 6 Index is, we need two fives in denominator, need 1 more
( ) 6 0 Multiply numerator and denominator by Example 8. 11 ( ) 11 88 Index is, we need four twos in denominator, need more Multiply numerator and denominator by Example 9. ( ) The can be written as. This will help us keep the numbers small Index is, we need three fives in denominator, need 1 more Multiply numerator and denominator by Multiply out denominator The previous example could have been solved by multiplying numerator and denominator by. However, this would have made the numbers very large and we would have needed to reduce our soultion at the end. This is why rewriting the radical as and multiplying by just was the better way to sim-
plify. We will also always want to reduce our fractions (inside and out of the radical) before we rationalize. Example. ( 11 6 1 1 11 ) 11 11 11 Reduce coefficients and inside radical Index is, need two elevens, need 1 more Multiply numerator and denominator by 11 Multiply denominator The same process can be used to rationalize fractions with variables. Example 11. 18 6x y z 8 xy 6 z 9 x y z ( ) 9 x y z y z y z 9 x y z yz 9 x y z 0yz Reduce coefficients and inside radical Index is. We need four of everything to rationalize, three more fives, two more y s and one more z. Multiply numerator and denominator by y z Multiply denominator Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/)
8. Practice - Multiply and Divide Radicals Multiply or Divide and Simplify. 1) 16 ) 1m 1m ) x x ) 6 ( +) 9) 1 ( +) 11) (n + ) 1) (+ )( + ) 1) ( )( 1) 1) ( a + a)( a + a) 19) ( )( ) 1) ) ) 1 0 1 6 ) 9) 1) x x y p p ) ) ) r 8r ) 1 ) r r 6) a a 8) ( + ) ) 1 ( +) 1) 1 ( v) 1) ( + )( + ) 16) ( + )( + ) 18) ( p + )( p + p) 0) ( 1)( m + ) ) ) 6) 1 1 8) 0) ) ) 6) 8) 1 xy 8n n 1 6 6 6m n Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/) 6
8. 1) 8 ) 6 ) 6m ) r r ) x x 6) 6a a ) + 6 8) + 9) 1 ) + 1 11) n + 1) 9 v 1) 1) 16 9 1) 1 11 16) 0 +8 + 1 + 1) 6a+a +6a 6 + a 1 18) p + 0 p 19) 6 + 0) m + + m 1) Answers - Multiply and Divide Radicals ) 1 ) 1 0 ) 1 ) 6) 1 ) 9 8) 9) xy 1y 0) x 16xy 1) 6p ) n ) ) ) 6) 1 8 8 8 ) r 8) n mn Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (http://creativecommons.org/licenses/by/.0/)