Radicals

Motivation:

                      In previous section, square root has been introduced.  We will check more on this.   Recall that

                                                  6² = 36

                                                  7² = 49

               But if something’s square is equal to 37,   how do we exactly represent this “something” ?  In previous section, we introduced the symbol  to stand for the positive solution for this question.  In general, we call this representation as “radical”.

 

 

Definition ( Radical )

             Let n be a natural number where n  2.

             When we use  ,  it means the following things

1.                (  )ⁿ = p

2.                When n is an even number,   represents non-negative number to satisfy the relationship above.  In this case, only when p  0 does  belong to the set of positive numbers.

3.                When n is an odd number,  represents the real number to satisfy the relationship in 1 .

 

             In general,  the symbol  is equivalent to p^1/n .

 

             When n=2,  we just omit the writing for “2” by using   when p  0.

 

Example:   =1.41421…

                   And  (  )² = 2 .   For the equation x² = 2,   x can be  , or -  .

                    is the positive solution to the equation x² = 2;  -  is the negative one.

 

 

Example:   =2

 

 

 

Example:     ( because 2³ = 8 )

 

 

 

Example:   = -2 

                   (-2)³ = -8;   that’s the only number we know in real numbers to satisfy that .

 

 

 

Simplifying Radicals

 

                  Let’s start with the following expression

                                                  .

                  By the definition of this symbol, we know that

                                            (  )⁴ = 2² = 4

                 Recall that we also have

                                      = p^1/n   ( from definition )

                 For exponent, we always have  (a^m)ⁿ = a^mn  .  In this case,

                                          = (2²)^1/4 =  2^1/2 =

 

                 Let’s verify if it satisfies the original expression:

                                      (  )⁴ = (2^1/2)⁴ = 2² = 4

 

                 It seems that there is no problem to claim   = .  Actually, this is from the commutative law of multiplication on exponent. We mention this when we introduce  that (a^m)ⁿ = a^mn  holds at least when m and n are rational numbers.  For rational numbers, you should always check if it can be reduced to the simplest form.  The situation is similar to radicals.   So,

 

                   If  g=gcd(a,b) and   a=gm   b = gn,  then

                                     =

 

Example:     =

 

Example:     =