"Combine the following sentences using the conjunctions given in the brackets"
Can I use "by using", "use" instead?
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using the conjunctions given in the brackets is an example of a participle clause. In particular, a participle clause with -ing implies that the action occurs at the same time as the action described by the main clause, e.g.:
They spent the rest of the journey in silence, looking at each other. (They were silent and at the same time looking at each other)
There's nothing preventing the main clause to be an imperative one, e.g.:
Open the door slowly, keeping the dog inside. (Open the door and at the same time don't let the dog go outside)
Your sentence is just another example of that:
Combine the following sentences using the conjunctions given in the brackets. (Combine the sentences and at the same time make sure you only use the conjunctions given in the brackets)
By added before the -ing form usually implies the manner in which some action is performed (by=by means of), e.g.:
By using magic, they managed to defeat the old dragon. (They managed to defeat the old dragon by means of magic)
Note that by can usually be omitted in this context as well, but sometimes it is necessary to avoid confusion:
The boy concluded the horses were still in the area by examining the hoofprints.
The boy concluded the horses were still in the area examining the hoofprints. (who examined the hoofprints, the boy or the horses?)
You could argue that your example also describes the manner in which the exercise is to be completed. Therefore, using by is not an error, though it makes the sentence a bit contrived.
As a final note, the only way use can be, well, used in your sentence is by splitting it into two separate instructions:
Combine the following sentences. Use the conjunctions given in the brackets.