Why doesn't the US make a nuclear powered space shuttle?

If the United States made a space shuttle that was nuclear powered then, theoretically speaking, wouldn't it have a virtually endless power supply letting it travel for as long as the half life of the material would allow? For example the half life of Uranium-238 is 4.5 Billions years, so given enough amount to power the shuttle at ridiculous speeds wouldn't a shuttle "theoretically" be able to continue going for 4.5 million years? I understand that harnessing the radioactive matter is an extremely difficult process and that making it safe to have all of this radioactive matter to attempt to take off and land safely without bringing a nuclear fallout upon ourselves would be difficult as well. If we can use nuclear power plants to power millions of homes why not use it for space exploration to other solar systems in our galaxy, and then maybe even other galaxies... maybe we could discover something more powerful and travel at light speed.