Solution

$$614125$$ can be factorized as:
$$614125=5\times 122825$$
$$=5\times 5\times 24565$$
$$=5\times 5\times 5\times 4913$$
$$=5\times 5\times 5\times 17\times 289$$
$$=5\times 5\times 5\times 17\times 17\times 17$$
Taking cube root on both the sides, we get
$$^3\sqrt {614125}=^3\sqrt {5\times 5\times 5\times 17\times 17\times 17}$$
$$=5\times 17$$
$$=85$$