Solution - Other Factorizations

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

Rearrange:

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

                     2^6*x^2^3-(2^9)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((26) • (x(2^3))) -  29  = 0 

Step  2  :

Equation at the end of step  2  :

  ((26) • (x2^3)) -  29  = 0 
 Step  3  :

Equation at the end of step  3  :

  (26 • x8) -  29  = 0 

Step  4  :

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   64x⁸ - 512  =   64 • (x⁸ - 8) 

Trying to factor as a Difference of Squares :

 5.2      Factoring:  x⁸ - 8 

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 : 8 is not a square !!

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

Polynomial Roots Calculator :

 5.3    Find roots (zeroes) of :       F(x) = x⁸ - 8
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  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  1  and the Trailing Constant is  -8.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  5  :

  64 • (x8 - 8)  = 0 

Step  6  :

Equations which are never true :

 6.1      Solve :    64   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 6.2      Solve  :    x⁸-8 = 0 

 Add  8  to both sides of the equation : 
                      x⁸ = 8
                     x  =  8th root of (8) 

 The equation has two real solutions  
 These solutions are  x = ± 8th root of 8 = ± 1.2968  
 

Two solutions were found :