Which is larger: $\left(\dfrac{1}{3}\right)^{4}$ or $\left(\dfrac{1}{4}\right)^{3}$?

Which is larger: $\left(\dfrac{1}{3}\right)^{4}$ or $\left(\dfrac{1}{4}\right)^{3}$?
  1. $\left(\dfrac{1}{3}\right)^{4}$
  2. $\left(\dfrac{1}{4}\right)^{3}$
  3. Equal
  4. Cannot determine

Solution

$\left(\dfrac{1}{3}\right)^{4} = \dfrac{1}{81}$ and $\left(\dfrac{1}{4}\right)^{3} = \dfrac{1}{64}$. Since $\dfrac{1}{64} > \dfrac{1}{81}$, $\left(\dfrac{1}{4}\right)^{3}$ is larger.

Asked in: IMO

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