8.5 Radicals - Rationalize Denominators

1 - Rationalize DenominatorsObjective: Rationalize the Denominators of radical is considered bad practice to have a radical in the denominator of a this happens we multiply the numerator and denominatorby the samething in order to clear the radical. In the lesson on dividingradicals we talkedabout how this was done with monomials. Here we will look at how this is donewith the binomial is in the numerator the process to Rationalize the denominator isessentially the same as with monomials. The only difference is we will have to dis-tribute in the 92 6 Want to clear6 in denominator,multiply by6 ( 3 9)2 6 (6 6 )We will distribute the6 through the numerator118 9 6 2 6 Simplify Radicals in numerator,multiply out denominator9 2 9 6 12 Take square root where possible3 2 9 6 12 Reduce by dividing each term by32 3 6 4 Our SolutionIt is important to remember that when reducing the fraction we cannot reducewith just the 3 and 12 or just the 9 and 12.

2 When we have additionor subtrac-tion in the numerator or denominator we must divide all termsby the problem can often be made easier if we first simplify any Radicals in 12x2 18x Simplify Radicals by finding perfect squares2 4 5x3 4 3x2 9 2x Simplify roots,divide exponents 2x25x 2x3 3 2x Multiply coefficients4x25x 2x3 3 2x Multiplying numerator and denominator by2x (4x25x 2x3 )3 2x (2x 2x )Distribute through numerator4x210x2 2x6x 3 2xSimplify roots in numerator,multiply coefficients in denominator4x310 2x6x 6xReduce,dividing each term by2x22x210 6x 3xOur SolutionAs we are rationalizing it will always be important to constantly check ourproblem to see if it can be simplified more. We ask ourselves, can the fraction bereduced? Can the Radicals be simplified? These steps may happen several timeson our way to the the binomial occurs in the denominator we will have to use adifferent strategyto clear the radical.

3 Consider23 5, if we were to multiply the denominator by3 we would have to distribute it and we would end up with3 5 3 . We havenot cleared the radical, only moved it to another part of the denominator. So ourcurrent method will not work. Instead we will use what is called a conjugate. Aconjugateis made up of the same terms, with the opposite sign in the for our example with3 5in the denominator, the conjugate would be3 +5. The advantage of a conjugate is when we multiply them together we have( 3 5)( 3 + 5), which is a sum and a difference. We know when we multiplythese we get a difference of squares. Squaring3 and 5, with subtraction in themiddle gives the product3 25= 22. Our answer when multiplying conjugateswill no longer have a square root. This is exactly what we 5 Multiply numerator and denominator by conjugate23 5(3 + 53 + 5)Distribute numerator,difference of squares in denominator2 3 +103 25 Simplify denoinator2 3 +10 22 Reduce by dividing all terms by 2 3 511 Our SolutionIn the previous example, we could have reduced by dividng by 2, giving the solu-tion3 + 5 11, both answers are 5 + 3 Multiply by conjugate,5 3 315 5 + 3 (5 3 5 3 )Distribute numerator,denominator is difference of squares75 45 5 3 Simplify Radicals in numerator,subtract in denominator25 3 9 5 2 Take square roots where possible5 3 3 5 2 Our SolutionExample 3x 4 5x3 Multiply by conjugate,4 + 5x3 2 3x 4 5x3 (4 + 5x3 4 + 5x3 )

4 Distribute numerator,denominator is difference of squares8 3x + 215x4 16 5x3 Simplify Radicals where possible8 3x + 2x215 16 5x3 Our SolutionThe same process can be used when there is a binomial in the numerator anddenominator. We just need to remember to FOIL out the 5 2 3 Multiply by conjugate,2 + 3 3 5 2 3 (2 + 3 2 + 3 )FOIL in numerator,denominator is difference of squares6 + 3 3 2 5 15 4 3 Simplify denominator6 + 3 3 2 5 15 1 Divide each term by16 + 3 3 2 5 15 Our Solution4 Example 5 3 7 5 6 + 4 2 Multiply by the conjugate,5 6 4 2 2 5 3 7 5 6 + 4 2 (5 6 4 2 5 6 4 2 )FOIL numerator,denominator is difference of squares10 30 810 15 42 +12 14 25 6 16 2 Multiply in denominator10 30 810 15 42 +12 14 150 32 Subtract in denominator10 30 810 15 42 +12 14 118 Our SolutionThe same process is used when we have variablesExample + 4x3 5x 3x Multiply by the conjugate,5x+ 3x 3x2x + 4x3 5x 3x (5x+ 3x 5x+ 3x )

5 FOIL in numerator,denominator is difference of squares15x22x + 3x6x2 + 5x4x3 +12x4 25x2 3xSimplify radicals15x22x + 3x26 +10x2x + 2x23 25x2 3xDivide each term byx15x2x + 3x6 +10x x + 2x3 25x 3 Our SolutionWorld View Note:During the 5th century BC in India, Aryabhata published atreatise on astronomy. His work included a method for findingthe square root ofnumbers that have many and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( ) Practice - Rationalize )4 + 2 3 9 3)4 + 2 3 5 4 5)2 5 5 413 7)2 3 3 3 9)53 5 + 2 11)25 + 2 13)34 3 3 15)43 + 5 17) 44 4 2 19)11 + 2 21)14 27 2 23)ab ab a 25)a+ab a +b 27)2 + 6 2 + 3 29)a b a+b 31)63 2 2 3 33)a ba b b a 35)2 5 3 + 5 2) 4 + 3 4 9 4)2 3 2216 6)5 + 4417 8)5 2 3 6 10)53 + 4 5 12)52 3 2 14)42 216)22 5 + 2 3 18)44 3 5 20)3 + 3 3 122)2 +10 2 + 5 24)14 7 14 + 7 26)a+ab a +b 28)2 5 + 3 1 3 30)a ba +b 32)aba b b a 34)4 2 + 33 2 + 3 36) 1 + 5 2 5 + 5 2 637)5 2 + 3 5 + 5 2 38)3 + 2 2 3 2 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License.

6 ( ) - Rationalize Denominators1)4 + 2 3 32) 4 + 3 123)2 + 3 54)3 145)213 565 526)85 + 417 687)6 938)30 2 3 189)155 5 2 4310) 5 3 +205 7711)10 2 2 2312)2 3 + 2 213) 12 9 3 1114) 2 2 415)3 5 16)5 3 217)1 + 2 18)163 + 4 5 4319)2 120)3 + 2 3 21)2 22)2 23)a 24)3 2 2 25)a 26)1327)4 2 3 + 2 6 3 2 28)2 5 215 + 3 + 3 229)a2 2a b +ba2 b30)a b 31)3 2 + 2 3 32)a b +b a a b33)a b +b a ab34)24 4 6 + 9 2 3 3 1535) 1 + 5 436)2 5 5 2 10+ 510 3037) 5 2 +10 3 + 6 538)8 + 3 6 10 Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative CommonsAttribution Unported License. ( )8