Solution:

Given:
a = ∛√2(1 - ∛2 - 1)

To find:
The value of a^3 3a-2

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Step 1: Simplifying the expression for 'a'

We are given that a = ∛√2(1 - ∛2 - 1)

Let's simplify this expression step by step.

1. Distribute the ∛√2 to each term inside the brackets:
a = ∛√2 - ∛√2∛2 - ∛√2

2. Simplify the expression ∛√2∛2:

Since ∛2 is the cube root of 2, we can rewrite it as 2^(1/3). Therefore,
∛√2∛2 = ∛√2 * 2^(1/3) = ∛(√2 * 2^(1/3)) = ∛(2^(1/2) * 2^(1/3))

Using the property of exponents, we can add the exponents when multiplying:
∛(2^(1/2) * 2^(1/3)) = ∛(2^((1/2) + (1/3))) = ∛(2^(5/6))

3. Simplify the expression ∛(2^(5/6)):

To simplify this expression, we can rewrite it as a fraction with a rational exponent:
∛(2^(5/6)) = (2^(5/6))^(1/3) = 2^((5/6) * (1/3)) = 2^(5/18)

Now, let's substitute this value back into the expression for 'a':
a = ∛√2 - 2^(5/18) - ∛√2

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Step 2: Evaluating the expression a^3 3a-2

The expression we need to evaluate is a^3 3a-2.

1. Substitute the value of 'a' we found in step 1:
a^3 3a-2 = (∛√2 - 2^(5/18) - ∛√2)^3 * 3(∛√2 - 2^(5/18) - ∛√2) - 2

2. Simplify the expression (∛√2 - 2^(5/18) - ∛√2)^3:

To simplify this expression, we can use the binomial expansion formula, which states that (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.

Applying this formula to our expression, we get:
(∛√2 - 2^(5/18) - ∛√2)^3 = (∛√2)^3 - 3(∛√2)^2(2^(5/18)) + 3(∛√2)(2^(5/18))^2 - (2^(5/18))^3

Simplifying each term separately:
(∛√2)^3 = (∛√2)(∛√2)(∛√2