Solution - Reducing fractions to their lowest terms

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "^-5" was replaced by "^(-5)". 1 more similar replacement(s)

Step  1  :

Equation at the end of step  1  :

                                                 b       
  (((16•(a3))•(b7))+((4•(a3))•(b2)))-((8a•———————————————)•(2a(-4)•b(-5)))
                                          ((8•(a3))•(b2))
 Step  2  :

Equation at the end of step  2  :

                                              b    
  (((16•(a3))•(b7))+((4•(a3))•(b2)))-((8a•—————————)•2a(-4)b(-5))
                                          (23a3•b2) 

Step  3  :

               b  
 Simplify   ——————
            23a3b2 

Dividing exponential expressions :

 3.1    b¹ divided by b² = b^{(1 - 2)} = b^(-1) = ¹/_b¹ = ¹/_b

Equation at the end of step  3  :

                                            1 
  (((16•(a3))•(b7))+((4•(a3))•(b2)))-((8a•————)•2a(-4)b(-5))
                                          8a3b
 Step  4  :

Equation at the end of step  4  :

                                 2 
  (((16•(a3))•(b7))+(22a3•b2))-————
                               a6b6
 Step  5  :

Equation at the end of step  5  :

                               2 
  ((24a3 • b7) +  22a3b2) -  ————
                             a6b6

Step  6  :

Rewriting the whole as an Equivalent Fraction :

 6.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  a⁶b⁶  as the denominator :

                       16a3b7 + 4a3b2      (16a3b7 + 4a3b2) • a6b6 
     16a3b7 + 4a3b2 =  ——————————————  =  ———————————————————————
                             1                     a6b6          

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

Step  7  :

Pulling out like terms :

 7.1     Pull out like factors :

   16a³b⁷ + 4a³b²  =   4a³b² • (4b⁵ + 1) 

Polynomial Roots Calculator :

 7.2    Find roots (zeroes) of :       F(b) = 4b⁵ + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  b  for which   F(b)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  b  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  4  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

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:

 4a3b2 • (4b5+1) • a6b6 - (2)      16a9b13 + 4a9b8 - 2 
 ————————————————————————————  =  ———————————————————
             a6b6                        a6b6        

Step  8  :

Pulling out like terms :

 8.1     Pull out like factors :

   16a⁹b¹³ + 4a⁹b⁸ - 2  =   2 • (8a⁹b¹³ + 2a⁹b⁸ - 1) 

Trying to factor a multi variable polynomial :

 8.2    Factoring    8a⁹b¹³ + 2a⁹b⁸ - 1 

Try to factor this multi-variable trinomial using trial and error 

 Factorization fails

Final result :

  2 • (8a9b13 + 2a9b8 + 1) 
  ————————————————————————
            a6b6