Question

Arătați că numărul a egal cu 4 la puterea 20 ori 3 plus 2 la puterea 40 ori 5 plus 2 la puterea 40 este pătrat perfect​

Answer 1

$a = {4}^{20} \times 3 + {2}^{40} \times 5 + {2}^{40} \\ = {( {2}^{2} )}^{20} \times 3+ {2}^{40} \times 5 + {2}^{40} \\ = {2}^{40} \times 3+ {2}^{40} \times 5 + {2}^{40} \\ = {2}^{40} (3 + 5 + 1) \\ = {2}^{40} \times 9 \\ = {( {2}^{20}) }^{2} \times {3}^{2} = {( {2}^{20} \times 3)}^{2}$

Answer 2

$4^{20}\cdot 3+2^{40}\cdot 5+2^{40}=(2^2)^{20}\cdot 3+2^{40}\cdot5+2^{40} =\\=2^{40}\cdot (3+5+1)=2^{40} \cdot 9=(2^{20}\cdot 3)^2 \\ \boxed{\bold{(2^{20}\cdot 3)^2 \to \ p\u{a}trat \ perfect}}$