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Evaluate the following expression:  $$2^{7}+2^{8}+\cdots +2^{19}+2^{20}$$ 
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$$1+2+\cdots +2^{20}=2^{21}-1$$  $$1+2+\cdots +2^{6} = 2^{7}-1$$  Subtracting the two equations we have:  $$2^{7}+2^{8}+\cdots +2^{20} = 2^{21}-2^{7}$$ 