Question

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Answer 1

Explicație pas cu pas:

(5-2V6)¹⁰⁰⁹×(V2+V3)²⁰¹⁸×(7-4V3)¹⁰⁰×(V3+2)

²⁰⁰

(5-2V6)¹⁰⁰⁹(5+2V6)¹⁰⁰⁹(7-4V3)¹⁰⁰×(7+4V3)¹⁰⁰

[(5-2V6)(5+2V6)]¹⁰⁰⁹[(7-4V3)(7+4V3)]¹⁰⁰

(25-24)¹⁰⁰⁹(49-48)¹⁰⁰=1¹⁰⁰⁹×1¹⁰⁰=1

Answer 2

$\displaystyle a=(5-2\sqrt{6} )^{1009}\cdot (\sqrt{2} +\sqrt{3} )^{2018}\cdot (7-4\sqrt{3} )^{100} \cdot (\sqrt{3} +2)^{200}\\ \\ a=(5-2\sqrt{6} )^{1009}\cdot((\sqrt{2} +\sqrt{3} )^2)^{1009}\cdot(7-4\sqrt{3} )^{100}\cdot ((\sqrt{3} +2)^2)^{100}\\ \\ a=(5-2\sqrt{6} )^{1009}\cdot ((\sqrt{2} )^2+2 \cdot \sqrt{2} \cdot \sqrt{3} +(\sqrt{3} )^2)^{1009}\cdot(7-4\sqrt{3} )^{100}\cdot \\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\cdot((\sqrt{3} )^2+2\cdot\sqrt{3} \cdot2+2^2)^{100}$

$a=(5-2\sqrt{6} )^{1009}\cdot(2+2\sqrt{6} +3)^{1009}\cdot(7-4\sqrt{3} )^{100}\cdot (3+4\sqrt{3} +4)^{100}\\ \\ a=(5-2\sqrt{6} )^{1009}\cdot(5+2\sqrt{6})^{1009} \cdot (7-4\sqrt{3} )^{100}\cdot (7+4\sqrt{3} )^{100 }\\ \\ a=((5-2\sqrt{6} )(5+2\sqrt{6}))^{1009} \cdot((7-4\sqrt{3} )(7+4\sqrt{3} ))^{100}\\ \\ a=(25-24)^{1009}\cdot (49-48)^{100}\\ \\ a=1^{1009}\cdot 1^{100}\\ \\ a=1 \cdot 1 \\ \\ a=1 \in\mathbb{N}$