Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "62.5" was replaced by "(625/10)".

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     (625/10)-(k^56)=0 

Step by step solution :

Step  1  :

            125
 Simplify   ———
             2 

Equation at the end of step  1  :

  125    
  ——— -  k56  = 0 
   2     

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  2  as the denominator :

           k56     k56 • 2
    k56 =  ———  =  ———————
            1         2   

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 :

 2.2       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:

 125 - (k56 • 2)     125 - 2k56
 ———————————————  =  ——————————
        2                2     

Trying to factor as a Difference of Squares :

 2.3      Factoring:  125 - 2k⁵⁶ 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  125  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Equation at the end of step  2  :

  125 - 2k56
  ——————————  = 0 
      2     

Step  3  :

When a fraction equals zero :

 3.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  125-2k56
  ———————— • 2 = 0 • 2
     2    

Now, on the left hand side, the  2  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   125-2k⁵⁶  = 0

Solving a Single Variable Equation :

 3.2      Solve  :    -2k⁵⁶+125 = 0 

 Subtract  125  from both sides of the equation : 
                      -2k⁵⁶ = -125
Multiply both sides of the equation by (-1) :  2k⁵⁶ = 125


Divide both sides of the equation by 2:
                     k⁵⁶ = 125/2 = 62.500
                     k  =  56th root of (125/2) 

 The equation has two real solutions  
 These solutions are  k = 56th root of (62.500) = ± 1.07664  
 

Two solutions were found :