  A. Dubickas and C. Smyth introduced the metric Mahler measure $$
  M_1(\alpha) = \inf\left\{\sum_{n=1}^N M(\alpha_n): N \in \mathbb N, \alpha_1
\cdots \alpha_N = \alpha\right\}, $$ where $M(\alpha)$ denotes the usual
(logarithmic) Mahler measure of $\alpha \in \overline{\mathbb Q}$. This
definition extends in a natural way to the $t$-metric Mahler measure by
replacing the sum with the usual $L_t$ norm of the vector $(M(\alpha_1), \dots,
M(\alpha_N))$ for any $t\geq 1$. For $\alpha \in \mathbb Q$, we prove that the
infimum in $M_t(\alpha)$ may be attained using only rational points,
establishing an earlier conjecture of the second author. We show that the
natural analogue of this result fails for general $\alpha\in\overline{\mathbb
Q}$ by giving an infinite family of quadratic counterexamples. As part of this
construction, we provide an explicit formula to compute $M_t(D^{1/k})$ for a
square-free $D \in \mathbb N$.
