Solution:

Given expression: a = (√2 + 1)^(1/3) - (√2 - 1)^(1/3)

To find the value of a^3 - 3a + 2, let's simplify the expression step by step.

Step 1: Simplifying the expression for a

Let's simplify the expression for a first.

a = (√2 + 1)^(1/3) - (√2 - 1)^(1/3)

We can use the formula for the difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)

Let's rewrite (√2 + 1)^(1/3) as a cube and (√2 - 1)^(1/3) as another cube.

Let p = (√2 + 1)^(1/3) and q = (√2 - 1)^(1/3)

Now, we can rewrite the expression for a as:

a = p - q

Step 2: Simplifying p and q

Let's simplify p first.

p = (√2 + 1)^(1/3)

To get rid of the cube root, we can raise both sides of the equation to the power of 3.

p^3 = (√2 + 1)

p^3 - 1 = √2

(p^3 - 1)^2 = (√2)^2

(p^3 - 1)^2 = 2

p^6 - 2p^3 + 1 = 2

p^6 - 2p^3 - 1 = 0

We can solve this equation to find the value of p^3.

Using the quadratic formula, let's assume x = p^3.

x^2 - 2x - 1 = 0

Solving this quadratic equation, we get two values of x:

x = 1 + √2 and x = 1 - √2

Since p^3 = x, we have two possibilities for p^3:

p^3 = 1 + √2 and p^3 = 1 - √2

Similarly, let's simplify q.

q = (√2 - 1)^(1/3)

Following the same steps as above, we can find two possibilities for q^3:

q^3 = 1 + √2 and q^3 = 1 - √2

Step 3: Simplifying a^3 - 3a + 2

Now that we have the values of p^3 and q^3, we can substitute them back into the expression for a.

a = p - q

Substituting the values of p^3 and q^3, we have two possible values for a:

a = (1 + √2)^(1/3) - (1 + √2)^(1/3) and a = (1 - √2)^(1/3) - (1 - √2)^(1/3)

Let's simplify each possibility separately:

For a = (1 + √2)^(1/3) - (1 + √2)^(1/3):

Let p = (1 + √2)^(1/3