Let's solve this step by step:

First, simplify the expression within the parentheses:
3 3/5 = 18/5
1 7/9 = 16/9

(18/5 - 16/9)
Find a common denominator for 5 and 9, which is 45. Then convert 18/5 to have a denominator of 45.
(18/5)*(9/9) = 162/45

(162/45 - 16/9)
Simplify the expression:
= 162/45 - 80/45
= 82/45

Next, simplify the expression within the second set of parentheses:
1/2 - 1/3
Find a common denominator for 2 and 3, which is 6.
(3/6 - 2/6) = 1/6

Now, substitute the simplified values back into the original expression:
52/45 - (82/45)*(1/6)

Perform the multiplication within the parentheses:
= 52/45 - 82/270
Find a common denominator for 45 and 270, which is 1350.
= 1176/1350 - 405/1350
= 771/1350

So, 52/45 - (3 3/5 - 1 7/9)*(1/2 - 1/3) simplifies to 771/1350.