Exponents & Roots
Exponents describe repeated multiplication. Roots reverse exponents. Used in CAGR calculations.
Practice This ConceptUnderstanding Exponents & Roots
An exponent tells you how many times to multiply a number by itself. 2³ = 2 × 2 × 2 = 8. A root does the reverse: the cube root of 8 is 2. In EPSO, you encounter exponents in CAGR (Compound Annual Growth Rate) — where growth compounds year after year. The nth root extracts the annual rate from a multi-year growth.
Formula
$CAGR = \left(\frac{V_f}{V_i}\right)^{\frac{1}{n}} - 1$
Key Rules
- x^n means multiply x by itself n times
- The nth root of x undoes x^n: if 2^3 = 8, then the cube root of 8 = 2
- x^(1/n) is the same as the nth root of x
- Compound growth is different from simple growth: 10% per year for 3 years ≠ 30% total
- On a calculator: use x^(1/n) for the nth root
Examples in Action
1
1.32^(1/4)
=
1.072
4th root of 1.32 → annual growth multiplier
2
1.072 - 1
=
0.072 → 7.2%
Subtract 1 to get the rate, multiply by 100
3
1.05^3
=
1.1576
5% growth for 3 years = 15.76% total, not 15%
Common Errors
- Using simple division (total% ÷ years) instead of the nth root
- Counting years wrong (2018 to 2022 is 4 years, not 5)
- Forgetting to subtract 1 after taking the root
Pro Tip
If a question mentions "annual growth rate" over multiple years, you MUST use the compound formula. Simple division always overestimates.