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.

Question

Tamara

Mathematics

12 July, 14:29

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.

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Answers (1)

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Seamus Hardy · 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