If 4b^2 + 1/b^2 = 2, then calculate the value of 18b^3 + 1/b^3, without solving for b

Given: 4b^2 + \frac{1}{b^2} = 2

We can square the given equation to get:
(4b^2 + \frac{1}{b^2})^2 = 2^2
Expanding both sides:
16b^4 + 2 + \frac{1}{b^4} = 4
16b^4 + \frac{1}{b^4} = 2

Now, we know that:
18b^3 + \frac{1}{b^3} = (4b^2) \cdot (4b^2 + \frac{1}{b^2}) - (18b^4 + \frac{1}{b^4})
Substitute the given values:
18b^3 + \frac{1}{b^3} = (4b^2) \cdot 2 - (16b^4 + \frac{1}{b^4})
Simplify further:
18b^3 + \frac{1}{b^3} = 8b^2 - 2

\boxed{18b^3 + \frac{1}{b^3} = 8b^2 - 2}