exponentially
exponentially — adverb
1. at an increasingly rapid rate where the amount multiplies instead of adding a fi
at an increasingly rapid rate where the amount multiplies instead of adding a fixed quantity, so growth speeds up dramatically over time
Omar saw his company's monthly sales grow exponentially in the first two years.
collocation: grow exponentially
The number of Jisoo's app users increased exponentially after it won a design award.
collocation: increase exponentially
Owen's after-school origami club grew exponentially as members taught their friends how to fold paper cranes.
Anong realized that computing power had risen exponentially while costs dropped at the same time.
- rapidly
describes speed but does not imply multiplication; more general and less intense
- dramatically
emphasizes the surprising or impressive nature of the change but not the mathematical pattern
- sharply
suggests a steep, sudden change rather than gradual multiplication over time
- gradually
implies slow, step-by-step change rather than accelerating multiplication
- marginally
describes a very small change, the opposite of the dramatic growth pattern
文法句型
exponentially + increase/grow/rise/expand
用法筆記
Frequently paired with verbs of increase such as grow, rise, expand, multiply, and improve. Not used with stative verbs (know, like, own). The emphasis is always on repeated multiplication, not steady addition.
常見錯誤
exponentially — adjective
- exponentiallypositive
- more exponentiallycomparative
- most exponentiallysuperlative
1. relating to the small raised number that tells you how many times to multiply a
relating to the small raised number that tells you how many times to multiply a base number by itself — for example, in 2³ the 3 is an exponential symbol showing 2 × 2 × 2
Kwame wrote exponential notation on the board to show that 5⁴ means 5 × 5 × 5 × 5.
example: exponential notation for repeated multiplication
Kemi wrote 7³ in her lab report to avoid repeating 7 × 7 × 7 on every line.
Madison used exponential symbols to shorten the repeated multiplication in her homework.
Ilan's calculator showed huge numbers in exponential format to save screen space.
用法筆記
Domain: mathematics. This sense refers specifically to the written notation of exponents (the small raised number). It is distinct from sense 3, which describes the function or growth pattern.
常見錯誤
2. describing a mathematical expression in which the exponent is a variable — such
describing a mathematical expression in which the exponent is a variable — such as 2ˣ or aⁿ — rather than a fixed number
Nikhil studied exponential equations where the unknown letter appeared in the exponent position.
subject + verb + exponential + noun: exponential equations
Kian saw that 3ˣ was an exponential expression with the variable in the exponent.
Eleni solved for x in the exponential equation 5ˣ = 125 by rewriting the base on both sides.
Ravindra stared at 2ⁿ = 64 on his algebra quiz and tried to find the value of n.
用法筆記
Domain: algebra. Unlike sense 1, which focuses on the notation, this sense is about expressions where the exponent itself is a variable. These are often taught alongside logarithmic functions.
3. relating to a mathematical function in which the rate of change is proportional
relating to a mathematical function in which the rate of change is proportional to the current value — representable by a formula such as f(x) = a·bˣ — producing multiplicative growth or decay over equal time periods
Vinícius drew an exponential curve showing bacteria doubling every hour.
collocation: exponential curve + concrete scenario (bacteria)
Eve modelled the spread of the virus using an exponential function with a base of 1.3.
Astrid watched the ant colony double every month and drew an exponential growth curve for her project.
Emily used an exponential function to calculate loan growth over ten years.
用法筆記
Domain: mathematics and sciences. This sense is the closest to the everyday meaning of exponential growth. Distinguish from sense 2: sense 2 focuses on expressions with a variable exponent, while this sense focuses on the functional relationship and its graph.