Exponents & Square Roots
Powers let you express repeated multiplication compactly. Roots undo them. Together they unlock scientific notation and beyond.
What Are Exponents?
An exponent tells you how many times to multiply a number by itself.
3⁴ = 3 × 3 × 3 × 3 = 81
The number (3) is the base. The exponent (4) is the power. Read as "3 to the fourth power."
Common Powers
Powers of 2
2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024
These show up constantly in computing (bytes, memory, etc.).
Powers of 3
3¹=3, 3²=9, 3³=27, 3⁴=81, 3⁵=243
Powers of 10
10¹=10, 10²=100, 10³=1,000, 10⁴=10,000, 10⁵=100,000, 10⁶=1,000,000
Pattern: 10ⁿ = 1 followed by n zeros. This is the basis of scientific notation!
Special Exponents
- x¹ = x (any number to the first power is itself)
- x⁰ = 1 (any non-zero number to the zero power is 1)
- x⁻¹ = 1/x (negative exponents create fractions)
- x⁻² = 1/x² (2⁻³ = 1/2³ = 1/8 = 0.125)
Exponent Rules
Rule 1: Multiplying Same Bases → Add Exponents
aᵐ × aⁿ = aᵐ⁺ⁿ
Example 1: 2³ × 2⁴ = 2⁷ = 128
Rule 2: Dividing Same Bases → Subtract Exponents
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Example 2: 5⁶ ÷ 5² = 5⁴ = 625
Rule 3: Power of a Power → Multiply Exponents
(aᵐ)ⁿ = aᵐˣⁿ
Example 3: (3²)³ = 3⁶ = 729
Rule 4: Power of a Product
(ab)ⁿ = aⁿ × bⁿ
Example 4: (2×5)³ = 2³ × 5³ = 8 × 125 = 1,000
Square Roots
The square root of a number asks: what number multiplied by itself gives this?
√25 = 5 because 5 × 5 = 25
√144 = 12 because 12 × 12 = 144
Perfect Squares to Know
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
(These are 1², 2², 3², ... 15²)
Cube Roots
The cube root asks: what number multiplied by itself THREE times gives this?
∛8 = 2 because 2 × 2 × 2 = 8
∛27 = 3 because 3 × 3 × 3 = 27
∛125 = 5 because 5 × 5 × 5 = 125
Scientific Notation (Introduction)
Scientific notation expresses very large or very small numbers using powers of 10:
a × 10ⁿ where 1 ≤ a < 10
Large Numbers
- 3,500,000 = 3.5 × 10⁶
- 92,000,000 = 9.2 × 10⁷
- Speed of light: 300,000,000 m/s = 3 × 10⁸ m/s
Small Numbers
- 0.005 = 5 × 10⁻³
- 0.00042 = 4.2 × 10⁻⁴
Count how many places you move the decimal. Right = negative exponent. Left = positive.
🎯 Try It Yourself
Test your understanding with these practice problems.
1. What is 2⁵?
💡 Hint: 2×2×2×2×2
2. What is √144?
💡 Hint: What number times itself equals 144?
3. What is 10⁴?
💡 Hint: 1 followed by 4 zeros
4. Simplify: 3² × 3³
💡 Hint: Add exponents: 3⁵ = 243
5. Express 45,000 in scientific notation
💡 Hint: Move decimal 4 places left: 4.5 × 10⁴
⚠️ Common Mistakes
- ✗Confusing 3² (which is 9) with 3×2 (which is 6). Exponents mean repeated multiplication, not regular multiplication.
- ✗Thinking x⁰ = 0. Actually, any non-zero number to the power of 0 equals 1. 5⁰ = 1.
- ✗Adding exponents when you should multiply: (3²)³ = 3⁶ = 729, NOT 3⁵ = 243.
- ✗Applying exponents to negative numbers incorrectly: (-3)² = 9 but -3² = -9 (the negative isn't squared).
- ✗Forgetting that square root of a negative number has no real solution: √(-4) is not a real number.
🌍 Real Life Example
Compound Interest Growth
If you invest $1,000 at 10% annual return, your money grows exponentially: Year 1: $1,000 × 1.1¹ = $1,100. Year 5: $1,000 × 1.1⁵ = $1,610.51. Year 20: $1,000 × 1.1²⁰ = $6,727.50. Year 30: $1,000 × 1.1³⁰ = $17,449.40. This is the power of exponential growth — and why starting to invest early matters so much.
💡 Key Takeaway
Exponents are shorthand for repeated multiplication: aⁿ means multiply a by itself n times. Square roots (√) are the inverse — they ask 'what number squared gives this?' Key rules: multiply bases → add exponents, power of a power → multiply exponents, anything to the 0 power = 1.