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This is an example of a double displacement reaction. The anion from the strong acid (usually fully dissociated in water) and the bicarbonate anion (along with it's cation) "switch" with each other (I know the wording is a bit crude..).
Take the commonly quoted example of hydrochloric acid and sodium bicarbonate.
HCl + NaHCO3 → NaCl + H2CO3
The salt, sodium chloride in this case, dissociates in the water, and the carbonic acid breaks apart into water and carbon dioxide. So really, the reaction is:
HCl + NaHCO3 → NaCl + H2O + CO2
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