  Let $(P_n)_{n\ge 0}$ and $(Q_n )_{n\ge 0}$ be the Pell and Pell-Lucas
sequences. Let $b$ be a positive integer such that $b\ge 2.$ In this paper, we
prove that the following two Diophantine equations $P_{n}=b^{d}P_{m}+Q_{k}$ and
$P_{n}=b^{d}Q_{m}+P_{k}$ with $d,$ the number of digits of $P_k$ or $Q_k$ in
base $b,$ have only finitely many solutions in nonegative integers $(m, n, k,
b, d)$. Also, we explicitly determine these solutions in cases $2 \le b \le
10.$
