Of all the carbon in the world only a very small amount is radioactive.
Plants are designed to absorb carbon dioxide (what you and I breathe out --.
But they omit discussion of the basic flaw in the method: you cannot measure the age of a rock using radioactive dating because you were not present to measure the radioactive elements when the rock formed and you did not monitor the way those elements changed over its entire geological history.
If you check this educational page by the US Geological Society you will see that they spend all their time talking about the technicalities of radioactive decay.
Carbon-14 is radioactive, with a half-life of about 5,700 years.