# Math Help - Naturel-number

1. ## Naturel-number

Prove :

$\left( {\sqrt[3]{{45 + 29\sqrt 2 }} + \sqrt[3]{{45 - 29\sqrt 2 }}} \right) \in {\rm N}$

2. Let $x=\sqrt[3]{45+29\sqrt{2}}+\sqrt[3]{45-29\sqrt{2}}$.

Then $x^3=\left(\sqrt[3]{45+29\sqrt{2}}+\sqrt[3]{45-29\sqrt{2}}\right)^3$.

For the right member I'll use the formula $(a+b)^3=a^3+b^3+3ab(a+b)$. But $a+b=x$ and $ab=\sqrt[3]{343}=7$

Then $x^3=90+21x\Rightarrow x^3-21x-90=0$.

The only real root, which is also integer is $x=6$.

3. Notice that (3 + sqrt 2)^3 = (45 + 29(sqrt 2)),

thus x = [3 + sqrt(2)] + [3 - sqrt (2)] = 3 + 3 = 6.