How does radioactive decay change an element?
To correct details in the previous responses: Transmutation is different from radioactive decay. The change in elemental composition defines the element number, not the other way around.
Radioactive DECAY changes an element by ejecting either an electron, proton, or alpha particle. electron particle type of the force of ejection determine whether it is a "beta" particle or a "gamma" particle.
Because an element is defined by the composition of its nucleus and electron shells, changing them results in a fundamental change in the type of element that it is. Changes in the electronic structure of an element are "normal" chemistry - that is how they react to form new compounds.
But, changes in the nuclear structure through "decay" and the ejection of protons from the nucleus, or pairs of nuclear protons and associated electrons, results in a different element, with corresponding different properties from the original.
Transmutation occurs by bombarding a nucleus with other elements or particles, building up (not decaying) new elements - although those new elements may themselves be "radioactive" and begin the sequence of decays to more stable states as well.