Question

Show that
\[\sqrt[3]{45+29\sqrt2}+\sqrt[3]{45-29\sqrt2}\]
is an integer.

Answer 1

Let k represent the above mentioned sum. Clearly, k is a real number. Now, consider the function f(x)=(x^3)-21x-90. Factoring this yields that f(x)=(x-6)[(x^2)+6x+15]. Hence, f(x)=0 has only one real solution, which also happens to be an integer. Finally it is seen that f(k)=0. Therefore k must be 6; an integer.

Answer 2

hello

Answer 3

looks like a solution to a cubic