The core of the Earth would still exist without the mantle or the crust. Due to the decrease in pressure it would expand a little. Without the insulating effect of the mantle, the remaining hot, liquid core would cool very quickly to a solid iron rich ball. The heat created by radioactive decay would probably not be enough to keep the outer core liquid. That heat would radiate quickly into space.
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OK, so you would hop up there like Batman, Steady your gait with your foot on Jupiter, Fly past Neptune, and place the core of our earth right in the middle of the next galaxy?
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Since its iron probably nothing