Dividing Radicals and Rationalizing the Denominator

Rationalizing the Denominator: In order for a radical expression to be in the simplest radical form,there can be no fractions under the radical sign and no radicals in the denominator of the fraction. Theprocess of removing the radicals from the denominator is called rationalizing the denominator.A. Rationalizing the denominator when the denominator has only one term with index nStep 1: Simplify any radicals in the numerator and the denominator.Step 2: Determine the factor needed to make the denominator radicand a perfect n th powerStep 3: Multiply the numerator and denominator with the factor determined in step 2Step 4: Simplify the resulting expression if possibleEx 5: Rationalize the denominatorStep 1: Simplify the radicalStep 2: The factor required to make the denominator radicand a perfect 2 nd power isStep 3: Multiplying the numerator and denominator by :Step 4: Simplify the resulting expression:Ex 6: Rationalize the denominatorStep 1: Simplify the radicalStep 2: The factor required to make the denominator radicand a perfect 3 rd power isStep 3: Multiplying the numerator and denominator by :Revised 04/10 3

Step 4: Simplify the resulting expression:Ex 7: Rationalize the denominatorStep 1: Simplify the radicalStep 2: The factor required to make the denominator radicand a perfect 3 rd power isStep 3: Multiplying the numerator and denominator by :Step 4: Simplify the resulting expression:Ex 8: Rationalize the denominatorStep 1: Simplify the radicalStep 2: The factor required to make the denominator radicand a perfect 3 rd power isStep 3: Multiplying the numerator and denominator by : =Step 4: Simplify the resulting expression:Revised 04/10 4

B. Rationalizing the denominator with two terms, one or both of which involve square root.Recall that the binomials and are called the conjugates. And the difference of squareformula the product of the conjugates will result in .Step 1: Multiply the numerator and denominator by the conjugate of the denominatorStep 2: Simplify the resulting expression if possibleEx 9: Rationalize the denominatorStep1: Multiplying the numerator and denominator with the conjugateof thedenominator:Step 2: Simplify:Ex 10: Rationalize the denominatorStep1: Multiplying the numerator and denominator with the conjugateof thedenominator:Step 2: Simplify:Ex 11: Rationalize the denominatorStep1: Multiplying the numerator and denominator with the conjugateof thedenominator:Step 2: Simplify:Revised 04/10 5

Exercises: Rationalize the denominator in each of the following. Assume that the variable arepositive and the denominator is equal to zero9. 10.11.13.12.14.15. 16.17.18.19. 20.21. 22.23.24.Solutions to the Exercise problems:Exercises: Divide and simplify using the following radical expressions1.= ==3.= ==2.= ===4.= ==Revised 04/10 6

5.= ==6.= ==7.==8.==9.==10.==11.=12.==13.==14.==Revised 04/10 7

15.==16.==17.==18.==19.==20.==21.==22.==23.==24.==Revised 04/10 8

You can get additional instruction and practice by going to the following websitehttp://www.purplemath.com/modules/radicals5.htm This website provides goodreview and practice problems for quotient rule and rationalizing the denominatorhttp://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut40_addrad.htm This website provides good review and practice for rationalizing thedenominatorhttp://www.helpalgebra.com/articles/rationalizedenominator.htm This websiteprovides good review and practice for rationalizing the denominatorRevised 04/10 9