Solution - Power equations

Step  1  :

 1.1   

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  

  1.2     6 = 2•3 (-6)⁵ = (2•3)⁵ = -2⁵ • 3⁵

Equation at the end of step  1  :

Step  2  :

 2.1   Negative number raised to an even power is positive 

For example let's look at (-7)⁶ , where (-7) , a negative number, is raised to 6 , an even exponent :

(-7)⁶ can be written as (-7)•(-7)•(-7)•(-7)•(-7)•(-7)

Now, using the rule that says minus times minus is plus, (-7)⁶ can be written as (49)•(49)•(49) which in turn can be written as (7)•(7)•(7)•(7)•(7)•(7) or 7⁶ which is positive.

We proved that (-7)⁶ is equal to (7)⁶ which is a positive number

Using the same arguments as above, replacing (-7) by any negative number, and replacing the exponent 6 by any even exponent, we proved which had to be proved

 2.2     6 = 2•3 (-6)⁸ = (2•3)⁸ = 2⁸ • 3⁸

Equation at the end of step  2  :

Step  3  :

             (28•38) 
 Simplify   —————————
            ( -25•35)

Dividing exponents :

 3.1    2⁸   divided by   2⁵   = 2^{(8 - 5)} = 2³

Dividing exponents :

 3.2    3⁸   divided by   3⁵   = 3^{(8 - 5)} = 3³

Final result :

  -216