# 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 \$\displaystyle x=2^{1/3}+3^{1/2}\$. Then for any positive integer n, the number \$\displaystyle x^n\$ is a linear combination with rational coefficients of some fixed numbers \$\displaystyle \{a_1,\dots,a_k\}\$. Find these \$\displaystyle a_k\$, and, in particular, k. This means that k + 1 numbers \$\displaystyle 1=x^0\$, \$\displaystyle x\$, \$\displaystyle x^2\$, ..., \$\displaystyle x^k\$ are linearly dependent over \$\displaystyle \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.