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 .
Accepted Solution
A:
x3-81=0 One solution was found : x = 3 • ∛3 = 4.3267Step 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)
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 rootsEquation 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