This is the 8876th most frequent Russian word.
применяя
"Applying."
The word 'применяя' is used here to describe the process by which 'new methods' are being utilized while solving 'tasks' or 'problems'.
Он решал задачи, применяя новые методы.
He solved problems by applying new methods.
In this sentence, 'применяя' indicates the action of using or utilizing the 'knowledge' obtained from the course to achieve a result.
Применяя знания из курса, студент смог написать сложную программу.
By applying knowledge from the course, the student was able to write a complex program.
Here 'применяя' highlights the approach or means of achieving 'productivity improvement' through the deployment of 'modern technologies'.