How to determine without calculator which is bigger, $\left(\frac{1}{2}\right)^{\frac{1}{3}}$ or $\left(\frac{1}{3}\right)^{\frac{1}{2}}$ How can you determine which one of these numbers is bigger (without calculating):
$\left(\frac{1}{2}\right)^{\frac{1}{3}}$ , $\left(\frac{1}{3}\right)^{\frac{1}{2}}$

Since $3^3=27 > 4=2^2$, we have
$$
\left(\frac{1}{2}\right)^2
>
\left(\frac{1}{3}\right)^3
$$
Take square roots and get
$$
\left(\frac{1}{2}\right)
>
\left(\frac{1}{3}\right)^{3/2}
$$
Take cube roots and get
$$
\left(\frac{1}{2}\right)^{1/3}
>
\left(\frac{1}{3}\right)^{1/2}
$$
