What will be the product when zinc and silver nitrate undergo a single-replacement reaction?

This single replacement reaction will produce silver metal, #"Ag"#, and aqueous zinc nitrate, #"Zn"("NO"_3)_2#.

Zinc is above silver is the metal reactivity series, so it will replace silver in silver nitrate.

The balanced chemical equation that describes this reaction looks like this

#"Zn"_ ((s)) + 2"AgNO"_ (3(aq)) -> 2"Ag"_ ((s)) + "Zn"("NO"_ 3)_ (2(aq))#

Because silver nitrate and zinc nitrate are both soluble ionic compounds, you can rewrite this equation as the complete ionic equation first

#"Zn"_ ((s)) + 2"Ag"_ ((aq))^(+) + 2"NO"_ (3(aq))^(-) -> 2"Ag"_ ((s)) + "Zn"_ ((aq))^(2+) + 2"NO"_ (3(aq))^(-)#

then remove the spectator ions, which are those ions that are present on both sides of the equation

#"Zn"_ ((s)) + 2"Ag"_ ((aq))^(+) + color(red)(cancel(color(black)(2"NO"_ (3(aq))^(-)))) -> 2"Ag"_ ((s)) + "Zn"_ ((aq))^(2+) + color(red)(cancel(color(black)(2"NO"_ (3(aq))^(-))))#

to get the net ionic equation

#"Zn"_ ((s)) + 2"Ag"_ ((aq))^(+) -> 2"Ag"_ ((s)) + "Zn"_ ((aq))^(2+)#

Notice that this is also a redox reaction.

Here zinc metal is being oxidized to zinc cations, #"Zn"^(2+)#, and the silver cations, #"Ag"^(+)#, are being reduced to silver metal.