Solution - Power equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-5" was replaced by "^(-5)".

Step  1  :

 1.1   Raising to a negative exponent

Raising a number to a negative exponent means 1 over that 
same number but now with a positive exponent.

For example, x(-5) is equal to 1/x(5) 


 1.2   

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  

  1.3     4 = 2² (-4)^-5 = (2²)^(-5) = (2)^(-10)

Equation at the end of step  1  :

  (-48) •  -(2)(-10)

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     4 = 2² (-4)⁸ = (2²)⁸ = 2¹⁶

Equation at the end of step  2  :

  216 •  -(2)(-10)

Step  3  :

Dividing exponents :

 3.1    2¹⁶   divided by   2¹⁰   = 2^{(16 - 10)} = 2⁶

Final result :

  -64