Solution - Simplifying radicals

Step by step solution :

Step  1  :

Polynomial Roots Calculator :

 1.1    Find roots (zeroes) of :       F(x) = x²+64
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  64.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2 ,4 ,8 ,16 ,32 ,64

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  x2 + 64  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    x²+64 = 0 

 Subtract  64  from both sides of the equation : 
                      x² = -64
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -64  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -64  =
                    √ -1• 64   =
                    √ -1 •√  64   =
                    i •  √ 64

Can  √ 64 be simplified ?

Yes!   The prime factorization of  64   is
   2•2•2•2•2•2 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 64   =  √ 2•2•2•2•2•2   =2•2•2•√ 1   =
                ±  8 • √ 1   =
                ±  8

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x=  0.0000 + 8.0000 i 
                      x=  0.0000 - 8.0000 i 

Two solutions were found :

1.   x=  0.0000 - 8.0000 i 
   
2.   x=  0.0000 + 8.0000 i