Charcoal, 30 bytes

ＮθＦθＦθ⊞υＸ⊕ι⁺²κＩ⟦⌈Φυ¬›ιθ⌊Φυ¬‹ιθ 

Try it online! Link is to verbose version of code. Explanation: Port of @Guiseppe's R answer.

Ｎθ 

Input n.

ＦθＦθ⊞υＸ⊕ι⁺²κ 

Generate a list of perfect powers from 1² to nⁿ⁺¹.

Ｉ⟦⌈Φυ¬›ιθ⌊Φυ¬‹ιθ 

Find the largest one not greater than n and the smallest one not less than n.

42 bytes for a less inefficient version:

ＮθＦＥ⊕₂θＸ⊕ι…·²⎇ιＬ↨θ⊕ι²Ｆι⊞υκＩ⟦⌈Φυ¬›ιθ⌊Φυ¬‹ιθ 

Try it online! Link is to verbose version of code. Explanation: As above but only loops up to ⌊√n⌋ and only powers up to 1+⌊logᵢn⌋ (or 2 if i=1).