Brachylog, 13 bytes
wm&l×ばつ
Function submission which dumps some garbage to STDOUT since it saves a byte over l~l&l×ばつ or ∧ċ&l×ばつ.
wm Create a list of unbound variables with the same length as the input.
(This also prints each element, but that's besides the point.)
<1 This list is strictly increasing,
ṗm and each of its elements is prime.
≜ Assign a value to each element.
;?z^m Raise each prime to the power of the corresponding element of the input,
×ばつ and output the product of the powers.
Without the ≜, it completely breaks given exponents of zero--I suppose because they are mathematically eliminated from the final product?
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