And to clarify, here's what you're missing :) :
You don't multiply each by 1000 to convert it to kg/m³.
There are 1,000 grams in 1 kg.
*But* there are 100x100x100 cubic centimetres in one cubic metre (one metre is 100 cm, right?), which is 1,000,000.
So, divide 7.9 g/m³ by 1,000 to get how many kilograms there are per cm³ (= 0.0079). Multiply 0.0079 by 1,000,000 to get how many kilograms there are per cubic metre, and you end up with 7,900 kg/m³ which is the correct answer.
Multiply that by 10 and you get a mass of 79,000 kg per 10 cubic metres.
If in doubt, just break it down step-by-step like this and the logic should be easier to follow :)
Hopefully that made things clearer?