To do it exactly we'd have to ask Shakuntala Devi! She'd be able to do it in a matter of seconds, but I don't know how. She wrote quite a few books on mental math, though.

I agree with Brent, we'd be best off estimating, but it'd still be tough. My rule of thumb is that (1 + small change) squared tends to be about 1 + 2*small change, so I'd work with that:

(1 + .035) * (1 + .035) ≈ 1.07

(1 + .07) * (1 + .07) ≈ 1.14

(1 + .14) * (1 + .14) ≈ 1.28

So 1.035 to the eighth gets us about 1.28. From there, we're looking at 1.28 (the eighth power) * 1.14 (the fourth power), so about 1.42. (The real answer is about 1.51, so you be the judge of whether this is close enough! The problem with this approximation is the more times you do it, the grosser the error becomes.)