Show that the value of $$\sqrt[3]{15+13\sqrt{2} } +\sqrt[3]{15-13\sqrt{2} } $$ is not an integer

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Show that the value of $$\sqrt[3]{15+13\sqrt{2} } +\sqrt[3]{15-13\sqrt{2} } $$ is  not an integer

Uniteasy Admin · Accepted Answer

Let
$$a = \sqrt[3]{15+13\sqrt{2} }$$
$$b=\sqrt[3]{15-13\sqrt{2} } $$
Then
$$ab = \sqrt[3]{15^2-13^2\cdot2}=-\sqrt[3]{113}$$
$$a^3+b^3 = 30$$
Apply sum of cubes identity
$$(a+b)(a^2-ab+b^2)$$
$$=(a+b)[(a+b)^2-3ab] = 30$$
Let $$x = a+b$$, then we get a cubic equation
$$x(x^2+3\sqrt[3]{113})=30$$
$$x^3+3\sqrt[3]{113}x-30=0$$
If $$x$$ is an integer, the equation will not hold true since multiplication or addition of an integer with a radical number will result in a radical number.
Therefore, $$x$$ is not an integer.

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