Exponential Growth
Next Lesson
Powers and Roots
Summary
Exponential patterns depend entirely on the base: powers can grow rapidly, shrink toward 0, or alternate signs while either increasing or decreasing in absolute value. The key ACT skill is to track direction and behavior rather than compute exact large powers.
- For a positive base greater than 1, successive powers increase and grow faster and faster, so exponential growth becomes enormous very quickly.
- For a positive base between 0 and 1, successive powers become smaller and smaller, showing that larger exponents can produce tinier values.
- For a negative base less than -1, the signs alternate between positive and negative while the absolute values increase, creating large jumps back and forth on the number line.
- For a negative base between -1 and 0, the signs also alternate, but the absolute values shrink, so the powers approach 0 by bouncing above and below it.
- Comparisons such as whether x^7 is greater than x^6 cannot be answered without knowing the type of base, but if x < 1 and x ≠ 0, then x^7 is never greater than x^6.
Chapters
00:00
Focus on Exponential Patterns Instead of Exact Values
00:44
Positive Bases: Growth Above 1 and Decay Between 0 and 1
02:59
Negative Bases: Alternating Signs with Growing or Shrinking Absolute Value
05:13
Using Base Type to Compare Powers Like x^7 and x^6