Question

[:vvv]

Giúp mình với ạ.

Answer 1

[Đặng Ngọc Quỳnh]

Ta có: 

\(a=\sqrt[3]{7+\sqrt{50}}=\sqrt[3]{7+5\sqrt{2}}=\sqrt[3]{\left(1+\sqrt{2}\right)^3}=1+\sqrt{2}\)

\(b=\sqrt[3]{7-\sqrt{50}}=\sqrt[3]{7-5\sqrt{2}}=\sqrt[3]{\left(1-\sqrt{2}\right)^3}=1-\sqrt{2}\)

\(\Rightarrow M=a+b=2\) là số chẵn (đpcm)

Lại có:

\(a+b=2;a.b=\left(1+\sqrt{2}\right).\left(1-\sqrt{2}\right)=-1\)\(;a^2+b^2=\left(a+b\right)^2-2ab=6\)

\(N=a^7+b^7\)

\(=\left(a^7+a^4b^3\right)+\left(b^7+a^3b^4\right)-\left(a^4b^3+a^3b^4\right)\)

\(=a^4\left(a^3+b^3\right)+b^4\left(a^3+b^3\right)-a^3b^3\left(a+b\right)\)

\(=\left(a^3+b^3\right)\left(a^4+b^4\right)+2\)

\(=\left(a+b\right)\left(a^2+b^2-ab\right)\left[\left(a^2+b^2\right)^2-2a^2b^2\right]+2\)

\(=2\left(7.34+1\right)=478\) là số chẵn(đpcm)