1. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.5 Denom Rationalize

2. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §7.4 → Add, Subtract, Divide Radicals  Any QUESTIONS About HomeWork §7.4 → HW-28 7.4 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 3 Bruce Mayer, PE Chabot College Mathematics Multiply Radicals  Radical expressions often contain factors that have more than one term.  Multiplying such expressions is similar to finding products of polynomials.  Some products will yield like radical terms, which we can now combine.

4. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 4 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Find the Product for  SOLUTION Use the distributive property. Multiply Using Product Rule for Radicals

5. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 5 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Find the Product for  SOLUTION (F.O.I.L.-like) Use the product rule. Use the distributive property. Find the products. Combine like radicals.

6. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Find the Product for  SOLUTION Simplify. Use (a – b) 2 = a 2 – 2ab + b 2

7. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Find the Product for  SOLUTION Simplify. Use (a + b)(a – b) = a 2 – b 2.

8. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Perform MultiTerm Multiplication  SOLUTION a) Using the distributive law

9. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Perform MultiTerm Multiplication  SOLUTION b) F O I L

10. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Multiply Radicals  Perform MultiTerm Multiplication  SOLUTION c) F O I L ( )

11. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 11 Bruce Mayer, PE Chabot College Mathematics Radical Conjugates  In part (c) of the last example, notice that the inner and outer products in F.O.I.L. are opposites, the result, m – n, is not itself a radical expression. Pairs of radical terms like, are called conjugates.

12. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 12 Bruce Mayer, PE Chabot College Mathematics Mult. Radicals by Special Prods  Multiplication of expressions that contain radicals is very similar to the multiplication of polynomials

13. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 13 Bruce Mayer, PE Chabot College Mathematics Mult. Radicals by Special Prods  Compare F.O.I.L. and Square of a BiNomial-Sum FOIL Method

14. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 14 Bruce Mayer, PE Chabot College Mathematics Rationalize DeNominator  When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator NO LONGER contains a RADICAL. The procedure for finding such an expression is called rationalizing the denominator.  We carry this out by multiplying by 1 in either of two ways.

15. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 15 Bruce Mayer, PE Chabot College Mathematics Rationalize → Method-1  One way is to multiply by 1 under the radical to make the denominator of the radicand a perfect power.  EXAMPLE  Rationalize Denom: Multiplying by 1 under the radical

16. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Rationalize DeNom  Rationalize DeNom:  SOLUTION Since the index is 3, we need 3 identical factors in the denom.

17. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 17 Bruce Mayer, PE Chabot College Mathematics Rationalize → Method-2  Another way to rationalize a DeNom is to multiply by 1 outside the radical.  EXAMPLE  Rationalize Denom: Multiplying by 1

18. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Rationalize DeNom  Rationalize DeNom:  SOLN Need in DeNom Radical

19. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Rationalize DeNom  Rationalize the denominator. Assume variables are >0  SOLN  Need in DeNom Radical 4 3 x 3

20. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 20 Bruce Mayer, PE Chabot College Mathematics Rationalize 2-Term Rad DeNoms  Recall that the Difference-of-2Sqs Product results in the O & I terms in the FOIL Multiplication Adding to Zero  To Rationalize a DeNominator that contains two Radical Terms requires the use of Conjugates (which have a Diff-of- Sqs form) to remove the radicals from the Denom

21. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 21 Bruce Mayer, PE Chabot College Mathematics Rationalize 2-Term Rad DeNoms  For Example to Rationalize the Denom of  Multiply the Numerator & Denominator by the CONJUGATE of the Original Denominator

22. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 22 Bruce Mayer, PE Chabot College Mathematics Example  Rationalize DeNom  Rationalize the denominator:  SOLUTION Multiplying by 1 using the conjugate

23. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example  Rationalize DeNom  Rationalize the denominator: Multiplying by 1 using the conjugate  SOLUTION

24. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 24 Bruce Mayer, PE Chabot College Mathematics Rationalize Numerator  To rationalize a numerator with more than one term, use the conjugate of the numerator  Example  Rationalize numerator  SOLUTION

25. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 25 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §7.5 Exercise Set 22, 38, 64, 74, 92, 128 → Derive φ  The Golden Ratio φ (phi)

26. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 26 Bruce Mayer, PE Chabot College Mathematics All Done for Today L. Da Vinci Used The Golden Ratio  Typo in Book for 1/GoldenRatio

27. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 27 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

28. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 28 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

29. BMayer@ChabotCollege.edu MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt 29 Bruce Mayer, PE Chabot College Mathematics