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Are there general rules for Base + Metal Compound reactions like there are for Acids?

For example, acids have some general rules.

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Acid + Metal => Salt + Hydrogen Gas

Acid + Metal Carbonate => Salt + Water + Carbon Dioxide

Acid + Metal Oxide => Salt + Water

Acid + Metal Hydroxide => Salt + Water

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Are there equivalent rules for bases?

Base + Metal => ?

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On the same level of generalization, the rule is "no reaction", but this rule has quite a few exceptions (probably more than the rules you quoted above).

Water-soluble bases would react with amphoteric metals to produce hydrogen and the corresponding salt: $$2\ce{Al + 2KOH + 6H2O \to 2KAl(OH)4 + 3H2}$$

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