Prove not

[MM]

Prove that any number of the form [tex]a\sqrt{b}+c\sqrt{d}[/tex] where [tex]a,b,c,d \in \mathbb{Z}[/tex] and [tex]b,d\ge0[/tex] is not transcendental.

[martin123456]

Prove that any number of the form [tex]a\sqrt{b}+c\sqrt{d}[/tex] where [tex]a,b,c,d \in \mathbb{Z}[/tex] and [tex]b,d\ge0[/tex] is not transcendental.

multiplying it by [tex]a\sqrt{b}-c\sqrt{d}[/tex] it gives [tex]a^2b-c^2d[/tex], summing it - [tex]2a\sqrt{b}[/tex]. So the expression is root of
[tex]x^2-2a\sqrt{b}+a^2b-c^2d=0[/tex]. now multiply LHS of this equation by
[tex]x^2+2a\sqrt{b}+a^2b-c^2d=0[/tex], so u get that [tex]a\sqrt{b}+c\sqrt{d}[/tex] is a root of the [tex](x^2+a^2b-c^2d)^2-4a^2b[/tex] that is with whole coefficients