Задача 39529 Найдите значение выражения...

slava191

2019-09-16 19:37:04

Найдите значение выражения [m]((\sqrt[4]{3}-\sqrt[4]{27})^2+7)((\sqrt[4]{3}+\sqrt[4]{27})^2-7)[/m]

sova

2019-09-16 20:11:32

★

Так как
[m](\sqrt[4]{3}-\sqrt[4]{27})^2=(\sqrt[4]{3})^2-2\sqrt[4]{3}\cdot\sqrt[4]{27} +(\sqrt[4]{27})^2=[/m]

[m]=\sqrt{3}-2\sqrt[4]{3\cdot 27}+\sqrt{27}=[/m]

[m]=\sqrt{3}-2\sqrt[4]{81}+3\sqrt{3}=4\sqrt{3}-2\cdot 3=4\sqrt{3}-6[/m]

и
[m](\sqrt[4]{3}+\sqrt[4]{27})^2=(\sqrt[4]{3})^2+2\sqrt[4]{3}\cdot\sqrt[4]{27} +(\sqrt[4]{27})^2=[/m]

[m]=\sqrt{3}+2\sqrt[4]{3\cdot 27}+\sqrt{27}=[/m]

[m]=\sqrt{3}+2\sqrt[4]{81}+3\sqrt{3}=4\sqrt{3}+2\cdot 3=4\sqrt{3}+6[/m]

то

[m]((\sqrt[4]{3}-\sqrt[4]{27})^2+7)\cdot ((\sqrt[4]{3}+\sqrt[4]{27})^2-7)=[/m]

[m]=(4\sqrt{3}-6+7)\cdot(4\sqrt{3}+6-7)=(4\sqrt{3}+1)\cdot(4\sqrt{3}-1)=[/m]

[m]=(4\sqrt{3})^2-1^2=48-1=47[/m]