Simplify the function f(x)=1/3(81)^3x/4 . then determine the key aspects of the function. the initial value is . the simplified base is . the domain is . the range is .

 1.1      Factoring:  x3-81 

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

 1.2    Find roots (zeroes) of :       F(x) = x3-81
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  -81. 

 The factor(s) are: 

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,3 ,9 ,27 ,81 

 Let us test

Polynomial Roots Calculator found no rational roots

 2.1      Solve  :    x3-81 = 0 

 Add  81  to both sides of the equation : 
                      x3 = 81 
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      x  =  ∛ 81  

 Can  ∛ 81 be simplified ?

Yes!   The prime factorization of  81   is
   3•3•3•3  
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).

∛ 81   =  ∛ 3•3•3•3   =
                3 • ∛ 3 

The equation has one real solution
This solution is  x = 3 • ∛3 = 4.3267 

step-by-step explanation:

mean or average is 40.6

correct answer is b. 27/8

step-by-step explanation: