Answer
$$=\frac{5 b^2}{ 9c^2}
.$$
Work Step by Step
We divide by the monomial as follows
$$
\frac{35a^3b^4c^6}{63a^3b^2c^8}=\frac{5\cdot 7a^3b^4c^6}{7\cdot 9a^3b^2c^8}=\frac{5 b^2}{ 9c^2}
.$$
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