📌 What Are Exponents?
An exponent shows how many times a number (called the base) is multiplied by itself.
Example: 2³ = 2 × 2 × 2 = 8
This is read as "2 raised to the power of 3."
🧮 Rules of Exponents
- Product Rule:
aⁿ × aᵐ = aⁿ⁺ᵐ - Quotient Rule:
aⁿ / aᵐ = aⁿ⁻ᵐ - Power of a Power:
(aⁿ)ᵐ = aⁿ×ᵐ - Power of a Product:
(ab)ⁿ = aⁿbⁿ - Zero Exponent:
a⁰ = 1(if a ≠ 0) - Negative Exponent:
a⁻ⁿ = 1 / aⁿ - Fractional Exponent:
a¹⁄ⁿ = ⁿ√a
🌱 Roots (Radicals)
The square root of a number is a value that, when multiplied by itself, gives the number.
Example: √9 = 3 because 3 × 3 = 9
Types of Roots
- Square root:
√x = x¹ᐟ² - Cube root:
∛x = x¹ᐟ³ - n-th root:
ⁿ√x = x¹ᐟⁿ
Roots and exponents are inverses of each other.
🔁 Exponents & Roots Together
(√a)² = aand(a²)½ = aaᵐ⁄ⁿ = ⁿ√(aᵐ)— Root and power combined
Example: 8²⁄³ = ∛(8²) = ∛64 = 4
🧩 Simplifying Expressions
You can combine exponent and root rules to simplify complex expressions:
(x²y³)⁴ = x⁸y¹²√(x⁶) = x³(if x ≥ 0)x⁻²y⁴ / x³y² = x⁻⁵y² = y² / x⁵
📈 Applications of Exponents & Roots
- Scientific notation:
5.3 × 10⁶ - Compound interest:
A = P(1 + r/n)ⁿᵗ - Physics: Square roots in motion formulas, energy equations
- Computer science: Logarithmic and exponential runtimes