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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 .

2 Answers

2 votes

Answer:

The initial value is

✔ 1/3

The simplified base is

✔ 27

The domain is

✔ all real numbers

The range is

✔ y > 0

Explanation:

User Roger Lehmann
by
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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)

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 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

One solution was found : x = 3 • ∛3 = 4.3267
User Jonathon Oates
by
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