There is $\root{\displaystyle{3}}\of{\frac{a}{b}}$ that will make the $3$ larger:

$$\root{\displaystyle{3}}\of{\frac{a}{b}}.$$

Don't know if this substantially better. If your custom is to use the \sqrt[#1]{#2} construction instead of my preferred (straight from the $\TeX$book) \root{#1}\of{#2} that probably works too.

Undoubtedly other finetuning is also possible.

There is $\root{\displaystyle{3}}\of{\frac{a}{b}}$ that will make the $3$ larger:

$$\root{\displaystyle{3}}\of{\frac{a}{b}}.$$

Don't know if this substantially better. If your custom is to use the \sqrt[#1]{#2} construction instead of my preferred (straight from the $\TeX$book) \root{#1}\of{#2} that probably works too.

Undoubtedly other finetuning is also possible.

Experimenting with the \raise[n pt]{} construct:

\root{\raise2pt{\displaystyle{3}}}\of{\frac{a}{b}}

$$\root{\raise2pt{\displaystyle{3}}}\of{\frac{a}{b}},$$

\root{\raise4pt{3}}\of{\frac{a}{b}}

$$\root{\raise4pt{3}}\of{\frac{a}{b}}.$$