Let's start by evaluating the arccosine of √3/2:

arccos(√3/2) = π/6

Next, let's evaluate the arccosine of √2/2:

arccos(√2/2) = π/4

Now, we can substitute these values back into the expression:

5arccos(√3/2) - 9arccos(√2/2)
= 5(π/6) - 9(π/4)
= (5π/6) - (9π/4)
= (10π/12) - (27π/12)
= (-17π/12)

Therefore, the final result is -17π/12.