Exponents and Square Roots
The rules of exponentiation, roots as inverse operations, and the patterns that make them useful for rapid calculation.
Exponents
An exponent tells you how many times to multiply a base by itself:
aⁿ = a × a × a × ... × a (n times)
- Base (a) — the number being multiplied
- Exponent / power (n) — how many times
2⁵ = 2 × 2 × 2 × 2 × 2 = 32
3⁴ = 81
10³ = 1000
The Laws of Exponents
These six rules cover all standard manipulations:
| Rule | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient | aᵐ / aⁿ = aᵐ⁻ⁿ | 2⁵ / 2² = 2³ = 8 |
| Power of power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of product | (ab)ⁿ = aⁿbⁿ | (2×3)² = 4×9 = 36 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Why a⁰ = 1: Using the quotient rule: aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰. And any number divided by itself is 1. So a⁰ = 1.
Why a⁻ⁿ = 1/aⁿ: Same logic: a² / a⁵ = a²⁻⁵ = a⁻³. And 1/a³ directly. So a⁻³ = 1/a³.
Fractional Exponents
Fractional exponents connect powers and roots:
a^(1/n) = ⁿ√a (nth root of a)
a^(m/n) = ⁿ√(aᵐ) (nth root of a to the mth power)
8^(1/3) = ∛8 = 2
27^(2/3) = (∛27)² = 3² = 9
4^(3/2) = (√4)³ = 2³ = 8
The denominator of the fractional exponent is the root; the numerator is the power. Order doesn’t matter — root first or power first gives the same result (take whichever is easier mentally).
Square Roots
The square root of a number n is the value that, when squared, gives n:
√n = x means x² = n
Every positive number has two square roots: positive and negative. √9 = ±3. By convention, √ denotes the positive root.
Perfect squares to memorise
| n | n² |
|---|---|
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 15 | 225 |
| 20 | 400 |
| 25 | 625 |
Simplifying square roots
Factor out perfect squares:
√72 = √(36 × 2) = 6√2
√200 = √(100 × 2) = 10√2
√48 = √(16 × 3) = 4√3
Key irrational values (approximate)
√2 ≈ 1.414
√3 ≈ 1.732
√5 ≈ 2.236
√10 ≈ 3.162
Trick: √2 ≈ 7/5 = 1.4 (good enough for estimation). √3 ≈ 7/4 = 1.75 (slightly over).
Powers of 2 and 10
These are the two sequences most worth having memorised cold.
Powers of 2
| n | 2ⁿ |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
| 5 | 32 |
| 6 | 64 |
| 7 | 128 |
| 8 | 256 |
| 9 | 512 |
| 10 | 1,024 ≈ 10³ |
| 20 | 1,048,576 ≈ 10⁶ |
| 30 | ≈ 10⁹ |
The approximation 2¹⁰ ≈ 10³ is extremely useful. Every 10 doublings ≈ 3 orders of magnitude.
Powers of 10
| n | 10ⁿ | Name |
|---|---|---|
| 3 | 1,000 | thousand |
| 6 | 1,000,000 | million |
| 9 | 10⁹ | billion |
| 12 | 10¹² | trillion |
| −3 | 0.001 | thousandth |
| −6 | 0.000001 | millionth |
Scientific Notation
A way to write very large or small numbers as a × 10ⁿ, where 1 ≤ a < 10.
3,700,000 = 3.7 × 10⁶
0.000045 = 4.5 × 10⁻⁵
Multiplication in scientific notation:
(3 × 10⁴) × (2 × 10³) = 6 × 10⁷
Multiply the coefficients, add the exponents. This is where the product rule pays off practically.
Estimating with Exponents
Rule of 72: a quantity growing at r% per period doubles in 72/r periods.
- 6% growth → doubles in 12 periods
- 10% growth → doubles in 7.2 periods
Compound growth in general:
Final = Initial × (growth factor)ⁿ
Small rates over many periods compound dramatically. 1.01¹⁰⁰ ≈ 2.7 (e). 1.07²⁰ ≈ 3.87. This is why exponents matter in finance, biology, and any compounding system.