Consider expressions built using number $1$, arithmetical operators $+, -, *, /$ and exponentiation ^ (in case of multiple values, the principal value is assumed, the same way as it implemented in Power function in Mathematica). Is it a decidable problem to check if such an expression is zero? If so, could you please point me to an algorithm that can solve this problem?

Update: I found a reference to Richardson's Theorem, that establishes undecidablity of equality in a wider set of expressions, in particular, including the logarithm and absolute value functions.