How was that calculated

http://www.wolframalpha.com/input/?i=%28%E2%88%9A3+%2B+%E2%88%9A2%29^9+*+%28%E2%88%9A3-%E2%88%9A2%29^9

Is it because one bracket of them is positive and the other is negative or what?

Powers can be distributed.

Here, you have
(a + b)^9 (a - b)^9

You can "un-distribute" the power:

[(a + b)(a - b)]^9

Solve the inside

[ a^2 - b^2 ]^9

In your problem,
a^2 = 3
b^2 = 2
a^2 - b^2 = 1
1^9 = 1

Just note the following equation:
(a + b)(a - b) = a² - b²

So:
(√3 + √2)⁹ * (√3-√2)⁹ = ( (√3 + √2) * (√3-√2) )⁹ = ( (√3)² - (√2)² )⁹ = (3 - 2)⁹ = 1⁹ = 1

(√3 + √2)^9 * (√3 - √2)^9

= [(√3 + √2)(√3 - √2)]^9

But √3 + √2)(√3 - √2) = 3 + √3√2 - √3√2 - 2 = 1

So you have 1^9 = 1