To clean buildup inside a vase, drop a couple denture cleanser tablets in the water with enough room to swish around. Leave it there over night and there should be nothing left but green water and a quick rinse should do the trick.
I use them to clean my toilet, too. They are amazing.
By Sandi/Poor But Proud from Salem, OR
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I find it hard to clean the bottom of some vases so I add a little warm water with an antacid tablet at the bottom. I let it sit overnight and just rinse. Works well.