I know the other way around is true, we can combine transcendental numbers to create algebraic ones, for example: $\pi/\pi=1$ or $\pi-\pi=0$. But what if $a$ and $b$ are algebraic, is there any combination of $a$ and $b$ that creates a transcedental one?
I'm aware of the theorem that states that $a^b$ is transcedental for algebraic $a \neq 0$ and irrational $b$. $\ln a$, $\sin a$ are also trans. numbers.
So i'm talking about adding, substraction, roots, multiplication, division. For example $a \pm b, \sqrt{a\pm b}, \sqrt{a}/b$ etc.
The operations you have provided are all algebraic; the sum of two algebraic numbers is algebriac, the difference, the product, the quotient (assuming you don't divide by 0), and the square root of two algebraic numbers are also algebraic.
Thus, any finite string you write involving algebraic numbers and these operations will also be algebraic.