# Prove that the number 2^(1/3)+3^(1/2) is algebraic

• Nov 19th 2013, 05:32 AM
peterty
Prove that the number 2^(1/3)+3^(1/2) is algebraic
Hey, i'm having trouble on how to tackle this question "Prove that the number 2^(1/3)+3^(1/2) is algebraic"
I know i have to show it as a polynomial but I don't know where to begin.
Any help would be appreciated

Thanks
• Nov 19th 2013, 07:56 AM
emakarov
Re: Prove that the number 2^(1/3)+3^(1/2) is algebraic
Let $x=2^{1/3}+3^{1/2}$. Then for any positive integer n, the number $x^n$ is a linear combination with rational coefficients of some fixed numbers $\{a_1,\dots,a_k\}$. Find these $a_k$, and, in particular, k. This means that k + 1 numbers $1=x^0$, $x$, $x^2$, ..., $x^k$ are linearly dependent over $\mathbb{Q}$. What does this mean?

Edit: I did not notice that this is a pre-university forum, so you may not know about linear dependence. Please say if the description above makes sense to you.