The value of $a^{1/2}\times a^{2/3}÷a^{1/6}$, when $a=64$, is:

Add the indices when multiplying and subtract when dividing: 1/2 + 2/3 − 1/6 = 3/6 + 4/6 − 1/6 = 1. So the expression is simply $a^1=64$. Adding the last index instead of subtracting would give $64^{4/3}=256$.