Naturel-number

**dhiab** · October 15th 2009, 12:04 AM

Prove :

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

**red_dog** · October 15th 2009, 01:44 AM

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 $x=6$ .

**pacman** · October 15th 2009, 06:50 AM

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

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

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