Question

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 .

Answer 1

 x3-81=0 One solution was found :                   x = 3 • ∛3 = 4.3267

Step by step solution :Step  1  :Trying to factor as a Difference of Cubes:

 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

Polynomial Roots Calculator :

 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 ....

  P  Q  P/Q  F(P/Q)   Divisor     -1     1      -1.00      -82.00        -3     1      -3.00      -108.00        -9     1      -9.00      -810.00        -27     1     -27.00     -19764.00        -81     1     -81.00     -531522.00        1     1      1.00      -80.00        3     1      3.00      -54.00        9     1      9.00      648.00        27     1      27.00     19602.00        81     1      81.00     531360.00   

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  : x3 - 81 = 0 Step  2  :Solving a Single Variable Equation :

 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 

One solution was found :                   x = 3 • ∛3 = 4.3267

Answer 2

X3-81=0 One solution was found : x = 3 • ∛3 = 4.3267
Step by step solution :Step 1 :Trying to factor as a Difference of Cubes:
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

Polynomial Roots Calculator :
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 ....
P Q P/Q F(P/Q) Divisor -1 1 -1.00 -82.00 -3 1 -3.00 -108.00 -9 1 -9.00 -810.00 -27 1 -27.00 -19764.00 -81 1 -81.00 -531522.00 1 1 1.00 -80.00 3 1 3.00 -54.00 9 1 9.00 648.00 27 1 27.00 19602.00 81 1 81.00 531360.00

Polynomial Roots Calculator found no rational roots
Equation at the end of step 1 : x3 - 81 = 0 Step 2 :Solving a Single Variable Equation :
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

One solution was found : x = 3 • ∛3 = 4.3267