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"improved exponentially" is correct and usable in written English
You can use it when you want to convey the idea that something has increased to a point far beyond the original level. For example, "His skills in mathematics have improved exponentially since he started studying with the tutor."
Exact(31)
Although our understanding of hematological malignancies has improved exponentially, resulting in greatly increased life expectancies and improved quality of life, our ability to abrogate tumor relapse has evaded us.
But over time, his work improved exponentially.
To his credit, he's improved exponentially in run defense.
"Our prospects have improved exponentially because the administration has given its stamp of approval," she said.
The fourth-year pro has improved exponentially since entering the league as a second-round pick.
Cooperatives, once notorious sources for poor wine, have improved exponentially, and viticulture and winemaking have moved into a new age.
Similar(29)
Historic trends suggest that finer resolution is all but assured; as complementary metal-oxide-semiconductor (CMOS) sensor technology continues to improve exponentially (Fossum 1997), resolution improvements will follow.
"Progress has to improve exponentially".
As a result, our knowledge of the late Mesozoic is improving exponentially, year on year.
When drones no longer need humans to control them, their usefulness will improve exponentially.
Their confidence improves exponentially and their performance at school improves," he said.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com