How to find out the greater number from $15^{1/20}$ and $20^{1/15}$?

$15<20, 15^{1/20}<20^{1/20}$ as $\dfrac1{20}>0$

Now, $20^{1/20}<20^{1/15}$ as $\dfrac1{20}<\dfrac1{15}$

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Alternatively, $$15^{1/20}<=>20^{1/15}\iff15^{15}<=>20^{20}$$

Now $15^{15}<20^{15}<20^{20}$

Well raise both numbers to the power of $20$

That is

$$\large{(15^\frac{1}{20})^{20} = 15^\frac{20}{20} = 15}$$

Now $$\large{(20^\frac{1}{15})^{20} = 20^\frac{20}{15} = 20^\frac{4}{3} = 20^{1.333..}}$$

which is greater ? $\large{15}$ or $\large{20^{1.333...}}$

Clearly , it is $\large{20^{1.333..}}$ because $\large{20^{1.333} > 20^1 > 15}$ and so this means that $\large{20^\frac{1}{15} >15^\frac{1}{20}}$