Rationalize the denominator of (3 * cube root of 12 - 8) / (cube root of 9 - cube root of 6).

Please show steps and the full process to rationalize.

(3∛12 - 8)/(∛9-∛6)

since a^3-b^3 = (a-b)(a^2+ab+b^2), multiply top and bottom by

(∛9^2 + ∛(9*6) + ∛6^2)

then you have

(∛36 - 6∛2 - 6∛3)/3

To rationalize the denominator of the given expression, we need to eliminate any cube roots in the denominator.

Step 1: Start by multiplying both the numerator and denominator by the conjugate of the denominator. The conjugate of the denominator is found by changing the sign of the term with the cube root of 6, so it becomes (cube root of 9 + cube root of 6).

((3 * cube root of 12 - 8) / (cube root of 9 - cube root of 6)) * ((cube root of 9 + cube root of 6) / (cube root of 9 + cube root of 6))

Step 2: Expand the numerator and denominator by multiplying using the FOIL method (First, Outer, Inner, Last).

((3 * cube root of 12 * cube root of 9) + (3 * cube root of 12 * cube root of 6) - (8 * cube root of 9) - (8 * cube root of 6)) / ((cube root of 9 * cube root of 9) - (cube root of 9 * cube root of 6) - (cube root of 6 * cube root of 9) + (cube root of 6 * cube root of 6))

Step 3: Simplify the expressions within the expanded numerator and denominator.

(3 * cube root of (12 * 9) + 3 * cube root of (12 * 6) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of (9 * 9) - cube root of (9 * 6) - cube root of (6 * 9) + cube root of (6 * 6))

(3 * cube root of 108 + 3 * cube root of 72 - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - cube root of 54 - cube root of 54 + cube root of 36)

Step 4: Simplify the cube roots.

(3 * cube root of (2 * 54) + 3 * cube root of (2 * 36) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - cube root of 54 - cube root of 54 + cube root of 36)

(3 * (cube root of 2) * cube root of 54 + 3 * (cube root of 2) * cube root of 36 - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - cube root of 54 - cube root of 54 + cube root of 36)

(3 * (cube root of 2) * 3 * (cube root of 18) + 3 * (cube root of 2) * 2 * (cube root of 18) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - cube root of 54 - cube root of 54 + cube root of 36)

Step 5: Simplify further if possible, by multiplying coefficients and combining like terms.

(9 * (cube root of 2) * (cube root of 18) + 6 * (cube root of 2) * (cube root of 18) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - 2 * cube root of 54 + cube root of 36)

Step 6: Simplify within the numerator and denominator.

(15 * (cube root of 2) + 6 * (cube root of 2) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - 2 * (cube root of 27) + cube root of 36)

(21 * (cube root of 2) - 8 * cube root of 9 - 8 * cube root of 6) / (cube root of 81 - 2 * (cube root of 27) + cube root of 36)

Step 7: Lastly, simplify the remaining cube roots if possible.

(21 * (cube root of 2) - 8 * (cube root of 9) - 8 * (cube root of 6)) / (cube root of 81 - 2 * (cube root of 3^3) + cube root of 36)

(21 * (cube root of 2) - 8 * (cube root of 9) - 8 * (cube root of 6)) / (cube root of 81 - 2 * 3 + cube root of 36)

(21 * (cube root of 2) - 8 * (cube root of 9) - 8 * (cube root of 6)) / (4 - cube root of 36)

This is the rationalized form of the given expression.

To rationalize the denominator of the given expression, we need to eliminate any cube roots in the denominator by multiplying the numerator and denominator by a suitable expression. Here are the steps to rationalize the denominator:

Step 1: Simplify the cube roots in the numerator and denominator:
- Cube root of 12 = 2√3 (since 2^3 = 8 and √3 is left inside the cube root)
- Cube root of 9 = √3 (since √3^3 = 27 and the cube root cancels the cube)
- Cube root of 6 remains unchanged.

So, the expression becomes:
(3 * 2√3 - 8) / (√3 - ∛ 6)

Step 2: Multiply the numerator and denominator by a suitable expression to eliminate the cube root in the denominator.
- The suitable expression in this case is (√3 + ∛6).

Multiply the numerator and denominator by (√3 + ∛6):

[(3 * 2√3 - 8) * (√3 + ∛6)] / [(√3 - ∛6) * (√3 + ∛6)]

Step 3: Simplify the expression in the numerator:
Distribute and combine like terms:

(6√3 + 6∛6 - 8√3 - 8∛6) / [(√3 - ∛6) * (√3 + ∛6)]

Simplifying further:
(-2√3 - 2∛6) / [(√3 - ∛6) * (√3 + ∛6)]

Step 4: Simplify the expression in the denominator:
In the denominator, we can use the difference of squares formula to simplify the expression (√3 - ∛6) * (√3 + ∛6):

[√3 * √3 + √3 * ∛6 - ∛6 * √3 - ∛6 * ∛6]
= [3 + √18 - √18 - 6]
= [-3]

So, the expression becomes:

(-2√3 - 2∛6) / (-3)

Step 5: Final simplification:
Divide the numerator by the denominator:

(-2√3 - 2∛6) / (-3)
= (2√3 + 2∛6) / 3

Therefore, the rationalized form of the given expression is (2√3 + 2∛6) / 3.