What keeps the earth's mantle and core hot?

Radioactivity. The three main radioactive elements in the mantle are U, K, and Th.

Three things are keeping Earth's core hot. !. The MASS ( pressure and volume)of the Earth's mantle and crust insulate the core and act much like multiple layers of blankets keep your body heat higher once the sheets around you have warmed up. I'm talking about NON-electric blankets and quilts. 2. As the pressure from rocks over lying the core increases, the temperature also rises and rocks and minerals melt the deeper they are because of the ideal and combined gas laws. PV = nRT 3. Radioactivity is contributing factor. but that is NOT the main reason why Earth's core is hot. Mars does not have enough mass to insulate the core, so mars core has lost enough heat for rocks to crystallize Mars core probably is not completely crystallized, yet, partly because of radioactivity., but that is why Mars no longer has GLOBAL magnetic filed like Earth has. Mars has REGIONAL magnetic fields which the remnant magnetic fields of iron nickel asteroids and meteorites impacting with enough force when Mars was forming AND after Mars formed that they are part of Mar's Mantle. . . Earth radiates more heat in the infrared out into space than it reflects from the Sun because the Earth's core is b the heat from the core is being conducted and convected through the mantel and crust.