E:15777 (Anca Mihiș, Baia Mare)
Arătaţi că numărul natural
![{\displaystyle A=\left(2^{2}\cdot 2^{4}\cdot 2^{6}\cdot \ldots \cdot 2^{2020}\right):\left(2\cdot 2^{3}\cdot 2^{5}\cdot \ldots \cdot 2^{2019}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb8cee34bfb835077ac800ebd5a5cceed55ee31)
este pătratul unui număr natural.
Soluția 1.
Folosind regulile de calcul cu puteri, numărul
devine
![{\displaystyle A=2^{2-1}\cdot 2^{4-3}\cdot 2^{6-5}\cdot \ldots \cdot 2^{2020-2019}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/699af8eff34a1a5c830fd73f7b9e4a8918b47d44)
Deci
![{\displaystyle A=2^{1010}=\left(2^{505}\right)^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89879484186d2fd80cac7ad57f344399cc064733)
aşadar
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
este pătratul perfect al numărului natural
![{\displaystyle 2^{505}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87417215acdac53450631bfef33a43872cc7ec21)
.
Soluția 2.
Deîmpărțitul este un produs de
factori, deci
![{\displaystyle 2^{2}\cdot 2^{4}\cdot 2^{6}\cdot \ldots \cdot 2^{2020}=2^{\frac {\left(2+2020\right)\cdot 1010}{2}}=2^{2022\cdot 505}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3783b07248805d4d06bd6aabd1bac49c0750c67)
Împărțitorul este un produs de
factori, deci
![{\displaystyle 2^{1}\cdot 2^{3}\cdot 2^{5}\cdot \ldots \cdot 2^{2019}=2^{\frac {\left(1+2019\right)\cdot 1010}{2}}=2^{2020}\cdot 505.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1789274f7cbe6a39383ee17ebd50bf681c1e0160)
Folosind regulile de calcul cu puteri, numărul
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
devine
![{\displaystyle A=2^{2022\cdot 505}:2^{2020\cdot 505}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84a4618ad3f03330c3c5e584c508538615767c86)
Cum
![{\displaystyle A=2022\cdot 505-2020\cdot 505=505\cdot \left(2022-2020\right)=505\cdot 2,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed08023c442c981f0599ea13b4feb08c84a2ed0)
avem că
![{\displaystyle A=2^{505\cdot 2}=\left(2^{505}\right)^{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3afe2812bbf611f28e09ddc20165962be39be1c7)
Deci
![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
este pătratul perfect al numărului natural
![{\displaystyle 2^{505}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87417215acdac53450631bfef33a43872cc7ec21)
.