Multiplying fractional exponents

Question

I'm doing Tim Roughgarden's Algorithms course and he has a slide with an integer multiplication algorithm.

Whats the rule that makes 10^(n/₂)a * 10^(n/₂)c become 10⁽ⁿ⁾ac ?

What do you do when multiplying fractional exponents like that?

Answer 1

It's based on the First Index Law, where:

a^m * aⁿ = a^{m + n}

in your case, the powers add to give ⁿ/₂ + ⁿ/₂ = ²ⁿ/₂ = n