Exponent – Fraction Exponent and Square Root

 

Motivation:

 

                        We have seen something like 2⁰ or 2^-1 in previous topic. Let’s explore more on it.   Consider the following expression

 

                             (2³)²

 

                      What does it mean ?  From the basic definition,

 

                             (2³)² = 2³  2³ = 2  2  2  2  2  2 = 2⁶

 

                      But what does it mean when fraction appears in exponent?  Like

 

                                2^1/2, 2^1/3, …

 

                      Similarly, if we hope the following rule still holds for fraction exponent, we have

                                (2^1/2)² = 2¹ = 2

 

                      In other words,   2^1/2 is one of the solutions of the following problem

 

                                           $² = 2

 

                      Since 2⁰ =1 and 2¹=2,  2^1/2 should sit between  1 and 2.  And that is also the reason why we would like to define the symbol for square root.

 

 

Definition ( Square Root )

                If  b  0,   we define

                             as the positive square root of  b.

 

              

                Why we say “positive square root” ?   The reason is

                                    (  )(  ) = b

                 And

                                    (-  )(-  ) = b

 

                  and -  are both square roots of b.

                And  is exactly b^1/2  .

 

 

 

 

 

                  In general, we  have the symbol  

                                                       ,  where n is an positive integer such that n>1 .

 

                  And it has the relationship

                                                     (  )ⁿ = p

 

                 While n = 2, we just omit the writing for 2.  And  is equivalent to  p^1/n .

 

Example: 

                   And  (  )³ = -2.    

 

                 We will introduce more on this while introducing radicals.  We just focus on fraction exponent here first.

 

                  Now, we consider  (a^p)^q  .

 

                  When p, q are integers,   we are pretty sure that  (a^p)^q = a^pq .

 

                  When p is fraction and q is integer,  the relation (a^p)^q = a^pq also holds.

 

                  What if p is integer and q is fraction?   Let’s take a look at it from simple example.

 

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

                            (2^1/2)²  = (  )² = 2

                  So,

                                (2²)^1/2   =   2^1/2)²   = 2 .     It seems that   (a^p)^q = a^pq also holds.

                   We can think in this way:  the commutative rule for multiplication should hold even when they appear on the exponent. In other words,  a^pq = a^qp .  So, if the relationship (a^p)^q = a^pq  holds when p is fraction and q is integer,  this relation should also hold when p is integer and q is fraction.  Otherwise, the commutative law for multiplication would not hold.

 

                   Thus, we have the following property or theorem.

 

 

 

Theorem:  (a^p)^q = a^pq ,  p, q are rational numbers.