Solution - Power equations

Step  1  :

            729
 Simplify   ———
            64 

Equation at the end of step  1  :

  729
  ———)1) ÷ 3
  64 

Step  2  :

 2.1   

Negative number raised to an odd power is negative

To show this we begin with (-7)⁵ ,a negative number raised to an odd exponent:

(-7)⁵ can be written as (-7)•(-7)•(-7)•(-7)•(-7)

Now, using the rule that says minus times minus is plus, (-7)⁵ can be written as (49)•(49)•(-7) which is a negative number. 

 We showed that (-7)⁵ is a negative number  

 Using the same arguments as above, replacing (-7) by any negative number, and replacing the exponent 5 by any odd exponent, we proved which had to be proved  

  2.2     729 = 3⁶ (-729)¹ = (3⁶)¹ = 3⁶  2.3     64 = 2⁶ (64)¹ = (2⁶)¹ = 2⁶

Equation at the end of step  2  :

  -36
  ——— ÷ 3
  26 

Step  3  :

         -36      
 Divide  ———  by  3
         26       

Dividing exponents :

 3.1    3⁶   divided by   3¹   = 3^{(6 - 1)} = 3⁵

Final result :

  -243            
  ———— = -3.79688 
   64