= Solution
Fix an <ordinal> $\rho>1$. The <Generalized Cantor normal form> asserts that every nonzero ordinal $\alpha$ has a unique expression
$$
\boxed{\alpha=\rho^{\beta_0}\xi_0+\rho^{\beta_1}\xi_1+\cdots+\rho^{\beta_{m-1}}\xi_{m-1}},
\qquad
\beta_0>\beta_1>\cdots>\beta_{m-1},\quad0<\xi_i<\rho.
$$
Products, sums and powers here are <ordinal multiplication>, <ordinal addition> and <ordinal exponentiation>; the coefficients need not be finite if $\rho$ is infinite. The case $\rho=\omega$ is <Cantor normal form>, and a finite base on finite ordinals gives ordinary positional notation. Zero has the empty expansion.
We first prove the <ordinal division algorithm>. For $\delta>0$, the function $q\mapsto\delta q$ is strictly increasing, continuous at limit ordinals and unbounded. Hence there is a greatest $q$ with $\delta q\le\alpha$: the first $q'$ with $\delta q'>\alpha$ cannot be a limit by continuity and must equal $q+1$. The interval after the initial segment $\delta q$ has a unique order type $r$, so $\alpha=\delta q+r$. Maximality of $q$ gives $r<\delta$. These facts also prove uniqueness of $q,r$.
Similarly $\beta\mapsto\rho^\beta$ is strictly increasing, continuous at limits and unbounded, so there is a unique largest $\beta_0$ with $\rho^{\beta_0}\le\alpha$. Divide by $\rho^{\beta_0}$ to obtain
$$
\alpha=\rho^{\beta_0}\xi_0+r_0,\qquad r_0<\rho^{\beta_0}.
$$
The quotient is positive, and $\alpha<\rho^{\beta_0+1}=\rho^{\beta_0}\rho$ makes $\xi_0<\rho$. If $r_0>0$, repeat on $r_0$; its leading exponent is strictly smaller than $\beta_0$. This process terminates, since an infinite run would give an infinite descending sequence of ordinals. It constructs the required form.
For uniqueness, a finite expression with largest exponent $\beta$ and coefficients below $\rho$ has value below $\rho^{\beta+1}$. Prove this from the end backwards: a tail with largest exponent $\gamma<\beta$ is below $\rho^{\gamma+1}\le\rho^\beta$, and
$$
\rho^\beta\xi+\text{tail}<\rho^\beta(\xi+1)\le\rho^\beta\rho.
$$
Thus the displayed leading exponent is the uniquely determined largest $\beta$ with $\rho^\beta\le\alpha$. The <ordinal division algorithm> uniquely determines its coefficient and the remaining tail, and recursion proves uniqueness of the whole expression. No commutative rearrangement of ordinal sums is used.
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