Solution - Power equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-3" was replaced by "^(-3)". 2 more similar replacement(s)

Step  1  :

 1.1     6 = 2•3

(6)^-3 = (2•3)^(-3) = (2)^(-3) • (3)^(-3)

Equation at the end of step  1  :

                     (-6-2)        
  (-6-4) +  ———————————————————————
            (0 - ((2)(-3)•(3)(-3))) 

Step  2  :

 2.1   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(-2) is equal to 1/x(2) 


 2.2   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

 2.3     6 = 2•3 (-6)^-2 = (2•3)^(-2) = (2)^(-2) • (3)^(-2)

Equation at the end of step  2  :

            2))-1 
  (-6-4) +  ———————
            (23•33)

Step  3  :

            1            -1  
 Divide  ———————  by  ———————
         (22•32)      (23•33)

 3.1    Dividing fractions

To divide fractions, write the divison as multiplication by the reciprocal of the divisor :

   1           -1            1        (23•33)
———————  ÷  ———————   =   ———————  •  ———————
(22•32)     (23•33)       (22•32)       -1   

Dividing exponents :

 3.2    2³   divided by   2²   = 2^{(3 - 2)} = 2¹ = 2

Raising to a Power :

 3.3    3³   divided by   3²   = 3^{(3 - 2)} = 3¹ = 3

Equation at the end of step  3  :

  (-6-4) +  -6

Step  4  :

 4.1   Negative number raised to an even power is positive 

 4.2     6 = 2•3

(-6)^-4 = (2•3)^(-4) = (2)^(-4) • (3)^(-4)

Equation at the end of step  4  :

  ((2)(-4)•(3)(-4)) +  -6

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  2⁴ • 3⁴  as the denominator :

          -6     -6 • (24•34)
    -6 =  ——  =  ————————————
          1        (24•34)   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 1 + -6 • 1296     -7775
 —————————————  =  —————
     1296          1296 

Final result :

  -7775            
  ————— = -5.99923 
  1296