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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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  1. 12 July, 14:45
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    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
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