Question

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Answer 1

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Step  1  :  Simplify —— x4 Equation at the end of step  1  : 1 1 1 1 (((x-—)•(x+—))•((x2)-————))•((x4)+——) 2 2 (x2) x4
 Step  2  :

Rewrite the whole as a fraction using  x4  as the denominator :

x4 x4 • x4 x4 = —— = ——————— 1 x4

Step2.2     :-       x4 • x4 + 1 x8 + 1 ——————————— = —————— x4 x4 Equation at the end of step  2  : 1 1 1 (x8+1) (((x-—)•(x+—))•((x2)-————))•—————— 2 2 (x2) x4
 Step  3    
Simplify —— x2 Equation at the end of step  3  : 1 1 1 (x8+1) (((x-—)•(x+—))•((x2)-——))•—————— 2 2 x2 x4
 Step  4.1   : 


Rewrite the whole as a fraction using  x2  as the denominator : x2 x2 • x2 x2 = —— = ——————— 1 x2
step4.2
Adding fractions that have a common denominator : 4.2       Adding up the two equivalent fractions 
 x2 • x2 - (1)     x4 - 1 —————————————  =  ——————      x2             x2  Equation at the end of step  4  :       1     1   (x4-1)  (x8+1)  (((x-—)•(x+—))•——————)•——————       2     2     x2      x4  sTEp 5       
    
Step  5  : 1 Simplify — 2 Equation at the end of step  5  : 1 1 (x4-1) (x8+1) (((x-—)•(x+—))•——————)•—————— 2 2 x2 x4
Step6:- 
Rewriting the whole as an Equivalent Fraction : 6.1   Adding a fraction to a whole 

Rewrite the whole as a fraction using  2  as the denominator : x x • 2 x = — = ————— 1 2 Adding fractions that have a common denominator : 6.2       Adding up the two equivalent fractions 
x • 2 + 1 2x + 1 ————————— = —————— 2 2 Equation at the end of step  6  : 1 (2x+1) (x4-1) (x8+1) (((x-—)•——————)•——————)•—————— 2 2 x2 x4 Step  7  : 1 Simplify — 2 Equation at the end of step  7  : 1 (2x + 1) (x4 - 1) (x8 + 1) (((x - —) • ————————) • ————————) • ———————— 2 2 x2 x4 Step  8  :Rewriting the whole as an Equivalent Fraction : 8.1   Subtracting a fraction from a whole 

Rewrite the whole as a fraction using  2  as the denominator : x x • 2 x = — = ————— 1 2 Adding fractions that have a common denominator : 8.2       Adding up the two equivalent fractions 
x • 2 - (1) 2x - 1 ——————————— = —————— 2 2 Equation at the end of step  8  : (2x - 1) (2x + 1) (x4 - 1) (x8 + 1) ((———————— • ————————) • ————————) • ———————— 2 2 x2 x4 Step  9  :Equation at the end of step  9  : (2x - 1) • (2x + 1) (x4 - 1) (x8 + 1) (——————————————————— • ————————) • ———————— 4 x2 x4


Trying to factor as a Difference of Squares : 10.1      Factoring:  x4-1 

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 : 1 is the square of 1
Check :  x4  is the square of  x2 

Factorization is :       (x2 + 1)  •  (x2 - 1) Trying to factor as a Difference of Squares : 10.3      Factoring:  x2 - 1 

Check : 1 is the square of 1
Check :  x2  is the square of  x1 

Factorization is :       (x + 1)  •  (x - 1) 
Equation at the end of step  10  : (2x-1)•(2x+1)•(x2+1)•(x+1)•(x-1) (x8+1) ————————————————————————————————•—————— 4x2 x4
______________

Multiplying exponential expressions : 11.2    x2 multiplied by x4 = x(2 + 4) = x6
Final result : (2x-1)•(2x+1)•(x2+1)•(x+1)•(x-1)•(x8+1) ——————————————————————————————————————— 4x6



Hope helped!