How did the Manchus, with a very small population in comparison to that of the Ming Dynasty, manage to conquer the great Chinese empire?

They took advantage of the chaos and anarchy plaguing the country

Explanation:

The Manchus succeeded as the ruling dynasty when the last Ming emperor died after the destruction of yellow river which unleashed the flood that destroyed hundreds of thousands of people coupled with the break down of social order, wide spread of smallpox and two people taking control of separate parts of the country and declaring new dynasties.