Solution - Power equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-3" was replaced by "^(-3)".

(2): "5.1" was replaced by "(51/10)". 2 more similar replacement(s)

Step  1  :

 1.1     10 = 2•5

(10)^-3 = (2•5)^(-3) = (2)^(-3) • (5)^(-3)

Equation at the end of step  1  :

   43         51
  (——•(104))+(——•((2)(-3)•(5)(-3)))
   10         10

Step  2  :

            51
 Simplify   ——
            10

Equation at the end of step  2  :

   43              51
  (—— • (104)) +  (—— • ((2)(-3)•(5)(-3)))
   10              10

Step  3  :

Multiplying exponents :

 3.1    2¹  multiplied by  2³   = 2^{(1 + 3)} = 2⁴

Raising to a Power :

 3.2    5¹  multiplied by  5³   = 5^{(1 + 3)} = 5⁴

Equation at the end of step  3  :

   43                51  
  (—— • (104)) +  ———————
   10             (24•54)

Step  4  :

 4.1     10 = 2•5

(10)⁴ = (2•5)⁴ = 2⁴ • 5⁴

Equation at the end of step  4  :

   43                  51  
  (—— • (24•54)) +  ———————
   10               (24•54)

Step  5  :

            43
 Simplify   ——
            10

Equation at the end of step  5  :

   43                  51  
  (—— • (24•54)) +  ———————
   10               (24•54)

Step  6  :

Dividing exponents :

 6.1    2⁴   divided by   2¹   = 2^{(4 - 1)} = 2³

Dividing exponents :

 6.2    5⁴   divided by   5¹   = 5^{(4 - 1)} = 5³

Equation at the end of step  6  :

                   51  
  (43•23•53) +  ———————
                (24•54)

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  2⁴ • 5⁴  as the denominator :

                   (43•23•53)     (43•23•53) • (24•54)
     (43•23•53) =  ——————————  =  ————————————————————
                       1                (24•54)       

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

Calculating Multipliers :

 7.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2⁴ • 5⁴

   Right_M = L.C.M / R_Deno = 1

Adding fractions that have a common denominator :

 7.3       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:

Multiplying exponents :

 2³  multiplied by  2⁴   = 2^{(3 + 4)} = 2⁷

Multiplying exponents :

 5³  multiplied by  5⁴   = 5^{(3 + 4)} = 5⁷ (43•2³•5³) • (2⁴•5⁴) + 51 43•2⁷•5⁷ + 51 ————————————————————————— = ————————————— (2⁴•5⁴) 1

Final result :

  43 + 51
  ———————
     1