Question

Вычислить $sqrt{15+2sqrt{26}}*sqrt{4+sqrt{1+2sqrt{26}}}*sqrt{4-sqrt{1+2sqrt{26}}}$

Объясните пожалуйста

Answer 1

$sqrt{15+2sqrt{26}}sqrt{4+sqrt{1+2sqrt{26}}}sqrt{4-sqrt{1+2sqrt{26}}}= \ =sqrt{15+2sqrt{26}}sqrt{(4+sqrt{1+2sqrt{26}})(4-sqrt{1+2sqrt{26}})}= \ =sqrt{15+2sqrt{26}}sqrt{16-1-2sqrt{26}}= \ =sqrt{15+2sqrt{26}}sqrt{15-2sqrt{26}}= \ =sqrt{(15+2sqrt{26})(15-2sqrt{26})}= \ =sqrt{225-4cdot26}=sqrt{225-104}=sqrt{121}=11$

Answer 2

$sqrt{15+2sqrt{26}}*sqrt{4+sqrt{1+2sqrt{26}}}*sqrt{4-sqrt{1+2sqrt{26}}}$
Последние два множителя можно подвести под общий корень(получим формулу вида a^2-b^2):
$sqrt{15+2sqrt{26}}*sqrt{(4+sqrt{1+2sqrt{26}})*(4-sqrt{1+2sqrt{26}})}=\=sqrt{15+2sqrt{26}}*sqrt{15-2sqrt{26}}$
И опять формула:
$sqrt{(15+2sqrt{26})*(15-2sqrt{26})}=sqrt{225-104}=sqrt{121}=11$