Let's break this down step by step:

Calculate cos(-2π/3):
cos(-2π/3) = cos(-120°) = cos(240°)

Using the unit circle, we know that cos(240°) = -1/2.
Therefore, cos(-2π/3) = -1/2.

Calculate tan(-3π/4):
tan(-3π/4) = tan(-135°)

Using the unit circle, we know that tan(-135°) = -1.
Therefore, tan(-3π/4) = -1.

Finally, multiply the results:
√3 cos(-2π/3) tan(-3π/4) = √3 (-1/2) (-1) = √3/2

Therefore, the final result is √3/2.