How is the core of the Earth kept hot?

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I know that the mantel of the Earth is kept hot by radioactivity, but is this also true for the core?
 
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Yes.

Addendum. Most of the radioactivity is in the crust and the mantle; but this is still generating heat which contributes to keeping the core hot. For lots more detail see Probing Question: What heats the Earth's core? at physorg.com

So the core is not being kept hot by radioactivity in the core; the energy from decay in the mantle and crust is more important. I had to look this up also.
 
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does the Earth gain significant heat from the moon and the suns tidal pull?
 
The link that sylas provided list 4 sources of heat within the earth.

Up to 90% of the heat is from radioactive decay.
About 5 to 10% is from friction caused by gravity.
About 5 to 10% is residual left over from the original formation of earth.
A small amount is latent heat released when molten materials turn into solids.

As I understand it, tidal push and pulls are part of the 5 to 10% gravity friction portion.