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					                                      Chapter 6 Cheat Sheet


Vocabulary:

Root:                                 Principal Root:


Radicand:                             Index:


Big Idea:         For every _______________ there is a _________________

Ex:        52 = 25, _________ is a sq. root
           53 = 125, _________ is a cube root
           55 = 3125, _________ is a fifth root


If an = b, then a is an nth root of b

How many different roots?

If n is:          Odd: one
                                                                      n
                  Even: and b is positive: 2 real roots (principal:       b and negative: - n b )
                        or if b is negative:   NO real roots

Note: If the problems says find ALL real roots, list principal AND negative
                                                                         
       If problems says find 4 x 2 , assume ONLY the principal (2x)

Examples:
                      


What are the roots?

                           n
For any real number a,         an =      a if n is __________
                                      Or |a| if n is ___________


Examples:       




More Problems: Pg. 364 1 – 6, 10-28 even, 33, 34, 42




                                                 Page 1
                                          Chapter 6 Cheat Sheet

     Steps to Simplify Radicals:
     A radical is in simplest form if all perfect nth factors are reduced

              1.   Does problem ask for all roots or only principal roots?
              2.   Find “a” – the root so that n a n (use prime factors, or guess and check)
              3.   Check the index
                   a. If index is ODD, n a n = a
                   b. If index is EVEN, n a n = |a|
                                    
              4.   Remove unnecessary absolute value signs:
                   a. If root exponent is EVEN, drop the absolute value symbol
                              
                   b. If root exponent is ODD, keep the absolute value
                                

     Examples:




     Combining Radical Expressions (Products):

          n           n                              n         n         n
     If       a and       b are real numbers, then       a *       b =       ab

     Note: because these are REAL numbers, a and b MUST be non-negative.
    
     Add to question 4 above:              
               c. If exponent in either radicand is ODD, drop the absolute value symbol

     Conditions for absolute value sign in final simplest form
     (look at each variable separately):
             Index is even
             Root exponent is odd
             Exponent in either original radicand is even

     Examples:




     More Problems: Pg. 371 10, 18, 19, 21, 29 – 31
     6-2 Review W/S Multiplying Radical Expressions


                                                         Page 2
                                          Chapter 6 Cheat Sheet

     Combining Radical Expressions (Quotients):

                                                     n
          n           n                                  a         a
     If       a and       b are real numbers, then   n
                                                           =   n
                                                         b         b


    
     Examples:
                                             




     Rationalizing the Denominator:
     Goal – no radicals in the denominator, and no denominator in any radical.

     Multiply numerator and denominator by a radical YOU CHOOSE. Make each
     component numbers that are perfect to the nth power.

     Examples:




     More Problems: Pg. 371 37 – 42
     6 -2 Review W/S, Multiplying and Dividing Radical Expressions



                                                         Page 3
                                   Chapter 6 Cheat Sheet

     Combining Radical Expressions: Sums and Differences

     Use the Distributive Property:
     You can combine LIKE radicals (have same index and radicand)

     an x  bn x  (a  b)n x

     Examples:





     Multiply Binomial Radical Expressions:
     Use mystery lots and combine like radicals

     Examples:




     Vocab:

     Conjugates:

     The product of two radical conjugates is a rational number.
     You can use this to rationalize a denominator.

     Examples:




     More Problems: Pg. 378 10-15, 17-22
                        6.1 – 6.3 Practice Worksheet




                                              Page 4
                                   Chapter 6 Cheat Sheet

 Section 6.4 – Rational Exponents
                              m
                  1
 Rule:   n
             a        AND a n  n am  (n a ) m
                  an

 Examples:
                  





 Things to Consider when simplifying radicals:
       Remember: Try and simplify radical first

 Same index AND same radicand:




 Same index, different radicand:




 Different index, same radicand:


                                              Page 5

				
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