**Solution:**

To find the value of (a - 1/a)^3, we first need to determine the value of a.

Given, a = 1 - √2

**Finding the value of a:**

To simplify the expression, let's rationalize the denominator of a by multiplying the numerator and denominator by the conjugate of the denominator:

a = (1 - √2)(√2 + 1)/(√2 + 1)(√2 + 1)
= (√2 + 1 - 2)/(2 - 1)
= (√2 - 1)/1
= √2 - 1

Therefore, the value of a is √2 - 1.

**Finding the value of (a - 1/a)^3:**

Now, let's substitute the value of a in the expression (a - 1/a)^3:

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

Let's simplify this expression step by step:

1. Multiply the first two terms together:
= (√2 - 1 - 1/√2 + 1)(√2 - 1 - 1/√2 + 1)
= (√2 - 1)^2 - (1/√2 - 1)^2

2. Expand the square of binomials:
= (√2 - 1)(√2 - 1) - (1/√2 - 1)(1/√2 - 1)
= 2 - 2√2 + 1 - (1/2 - √2 + 1)(1/2 - √2 + 1)
= 3 - 2√2 - (1/4 - √2/2 + √2/2 - 1)
= 3 - 2√2 - 1/4 + √2/2 - √2/2 + 1

3. Combine like terms:
= 4 - 1/4
= 15/4

Therefore, the value of (a - 1/a)^3 is 15/4.

20+ 14√2