#### Question

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

Question

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

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.