How do you put $\sqrt{3}\sqrt[3]{6}$ into rational exponents?

Question

For my precalc class we had the problem $\sqrt 3\sqrt[3]6$ and we had to convert it to rational exponents. I thought it was easy, $3^\frac 12\times6^\frac 13$ but the answer was $2^\frac 13\times3^\frac 56$. Any explanation please?

Answer 1

Your answer is correct, but it could be simplified further: $$3^\frac{1}{2}\times6^\frac{1}{3}=3^\frac{1}{2}\times2^\frac{1}{3}\times3^\frac{1}{3}=2^\frac{1}{3}\times3^\frac{5}{6}$$

Answer 2

As answered before, many equivalent solutions can be found. So you should try to put yourself in your teacher's shoes: gear toward "uniqueness" and simplicy. The $2^{\frac{1}{3}}3^{\frac{5}{6}}$ is somehow simpler because the factors are primes, and are written in increasing order. So you gain in simplicy (in a teacher's point of view) by careful factoring. Your initial guess, though correct, is a mere translation of the root expression, and brings no "added value" to your reasoning.