Question

Find the value of 729/4096 -1/3

Answer 1

Answer:

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

                    k^3-(729/4096)=0  

Step by step solution :

Step  1  :

            729

Simplify   ————

           4096

Equation at the end of step  1  :

          729

 (k3) -  ————  = 0  

         4096

Step  2  :

Rewriting the whole as an Equivalent Fraction :

2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  4096  as the denominator :

          k3     k3 • 4096

    k3 =  ——  =  —————————

          1        4096    

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:

k3 • 4096 - (729)     4096k3 - 729

—————————————————  =  ————————————

      4096                4096    

Trying to factor as a Difference of Cubes:

2.3      Factoring:  4096k3 - 729  

Theory : A difference of two perfect cubes,  a3 - b3 can be factored into

             (a-b) • (a2 +ab +b2)

Proof :  (a-b)•(a2+ab+b2) =

           a3+a2b+ab2-ba2-b2a-b3 =

           a3+(a2b-ba2)+(ab2-b2a)-b3 =

           a3+0+0+b3 =

           a3+b3

Check :  4096  is the cube of  16  

Check :  729  is the cube of   9  

Check :  k3 is the cube of   k1

Factorization is :

            (16k - 9)  •  (256k2 + 144k + 81)

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