Exponent – Negative Exponent

 

Motivation:

                        Initially for simplicity reason, we represent 

           

                                          2  2  2 = 2³     .

 

                        This notation saves us some time to write down lengthy expression if we have multiply a value by 2 one hundred times :

 

                                       7  2¹⁰⁰ = 7  2  2  2  …..  2     

                                                         (writing 2 for 100 times )

 

                       Furthermore, we have

 

                                    2³  2⁵ = 2⁽³⁺⁵⁾ = 2⁸

 

                       Naturally, we have this result because multiplying something by 2 three times, and then multiplying that result by 2 five times is equivalent to multiply by 2 for eight times.   But let’s consider more.  What is the meaning of the following expression

 

                                   2⁰ ,  2^-3 ????

 

                       When we are dealing with something new,  we always check the result we established previously.  In this case, we hope that  the following relation can hold

 

                                  2^b  2^c = 2^(b+c)

 

                       With this in mind,  we have

 

                                   2³ = 2⁽⁰⁺³⁾ = 2⁰  2³

 

                        Thus,  in order to have that relation holds without inventing new rules, we have

                                  2⁰ = 1

 

                      Similarly,

 

                                   1=2⁰ = 2^(3+(-3)) = 2³  2^-3

 

                     In other words,

                                                 

                                             1 = 2³  2^(-3)

 

 

 

                     So,  we have

 

                                          2^-3 =

 

                      In general, we have the following property

 

Property:

                  For  d  0 ,  p, q : integers

 

                     1.       d^p+q = d^p  d^q

                     2.       d⁰ = 1

                     3.       d^-p = 1/(d^p)