Solution - Simplifying radicals

Rearrange:

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

                     5*x^3/4-(40)=0 

Step by step solution :

Step  1  :

            x3
 Simplify   ——
            4 

Equation at the end of step  1  :

       x3     
  (5 • ——) -  40  = 0 
       4      

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

          40     40 • 4
    40 =  ——  =  ——————
          1        4   

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:

 5x3 - (40 • 4)     5x3 - 160
 ——————————————  =  —————————
       4                4    

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   5x³ - 160  =   5 • (x³ - 32) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  x³ - 32 

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 :  32  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = x³ - 32
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  -32.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  5 • (x3 - 32)
  —————————————  = 0 
        4      

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  5•(x3-32)
  ————————— • 4 = 0 • 4
      4    

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   5  •  (x³-32)  = 0

Equations which are never true :

 4.2      Solve :    5   =  0

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

Solving a Single Variable Equation :

 4.3      Solve  :    x³-32 = 0 

 Add  32  to both sides of the equation : 
                      x³ = 32
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 32  

 Can  ∛ 32 be simplified ?

Yes!   The prime factorization of  32   is
   2•2•2•2•2 
To be able to remove something from under the radical, there have to be  3  instances of it (because we are taking a cube i.e. cube root).

∛ 32   =  ∛ 2•2•2•2•2   =
                2 • ∛ 4

The equation has one real solution
This solution is  x = 2 • ∛4 = 3.1748

One solution was found :