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Question Number 125862 by mathdave last updated on 14/Dec/20

Answered by bramlexs22 last updated on 14/Dec/20

(√((7^(2012) (49−1))/(12))) = (√(7^(2012) ×4)) = 2×7^(1006) = a×7^b a=2 ∧ b=1006

$$\sqrt{\frac{\mathrm{7}^{\mathrm{2012}} \left(\mathrm{49}−\mathrm{1}\right)}{\mathrm{12}}}\:=\:\sqrt{\mathrm{7}^{\mathrm{2012}} ×\mathrm{4}} \\ $$$$\:=\:\mathrm{2}×\mathrm{7}^{\mathrm{1006}} \:=\:{a}×\mathrm{7}^{{b}} \\ $$$${a}=\mathrm{2}\:\wedge\:{b}=\mathrm{1006} \\ $$

Answered by talminator2856791 last updated on 15/Dec/20

(√((7^(2014) −7^(2012) )/(12))) = a(7^b ) (√((7^(2014) −7^(2012) )/(12))) = (√((49(7^(2012) )−7^(2012) )/(12))) = (√((7^(2012) (49−1))/(12))) = (√((7^(2012) ×48)/(12))) = (√((7^(2012) ×12×4)/(12))) = (√(4×7^(2012) )) = 2×7^(1006) = 2(7^(1006) ) a = 2(7^k ) b = 7^(1006−k) , k ∈ N

$$\:\sqrt{\frac{\mathrm{7}^{\mathrm{2014}} −\mathrm{7}^{\mathrm{2012}} }{\mathrm{12}}}\:=\:{a}\left(\mathrm{7}^{{b}} \right) \\ $$$$\:\sqrt{\frac{\mathrm{7}^{\mathrm{2014}} −\mathrm{7}^{\mathrm{2012}} }{\mathrm{12}}}\:=\:\sqrt{\frac{\mathrm{49}\left(\mathrm{7}^{\mathrm{2012}} \right)−\mathrm{7}^{\mathrm{2012}} }{\mathrm{12}}} \\ $$$$\:=\:\sqrt{\frac{\mathrm{7}^{\mathrm{2012}} \left(\mathrm{49}−\mathrm{1}\right)}{\mathrm{12}}}\:=\:\sqrt{\frac{\mathrm{7}^{\mathrm{2012}} ×\mathrm{48}}{\mathrm{12}}} \\ $$$$\:=\:\sqrt{\frac{\mathrm{7}^{\mathrm{2012}} ×\mathrm{12}×\mathrm{4}}{\mathrm{12}}}\:=\:\sqrt{\mathrm{4}×\mathrm{7}^{\mathrm{2012}} } \\ $$$$\:=\:\mathrm{2}×\mathrm{7}^{\mathrm{1006}} \:=\:\mathrm{2}\left(\mathrm{7}^{\mathrm{1006}} \right) \\ $$$$\:{a}\:=\:\mathrm{2}\left(\mathrm{7}^{{k}} \right) \\ $$$$\:{b}\:=\:\mathrm{7}^{\mathrm{1006}−{k}} \:,\:{k}\:\in\:\mathbb{N}\: \\ $$$$\: \\ $$

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