Basic Exponents

1. What is an Exponent?

An exponent tells us how many times a number (called the base) is multiplied by itself.

2. Understanding Exponential Notation

Instead of writing repeated multiplication, we use exponents.

3. Special Exponent Rules

Rule 1: Any Number to the Power of 1
Any number raised to the power of 1 is itself: a¹ = a
Example: 7¹ = 7 and (-3)¹ = -3

Rule 2: Any Number to the Power of 0
Any number (except 0) raised to the power of 0 is 1: a⁰ = 1
Example: 10⁰ = 1, (-5)⁰ = 1, 1000⁰ = 1

Rule 3: Multiplying Powers with the Same Base
a^m × aⁿ = a^m+n
Example: 2³ × 2⁴ = 2⁷ = 128 and x⁵ × x² = x⁷

Rule 4: Dividing Powers with the Same Base
a^m ÷ aⁿ = a^m-n
Example: 5⁶ ÷ 5² = 5⁴ = 625 and x⁹ ÷ x³ = x⁶

Rule 5: Power of a Power Rule
(a^m)ⁿ = a^m×n
Example: (3²)³ = 3⁶ = 729 and (x⁴)² = x⁸

Rule 6: Power of a Product Rule
(ab)ⁿ = aⁿ × bⁿ
Example: (2×5)³ = 2³ × 5³ = 8 × 125 = 1000

Rule 7: Power of a Fraction Rule
When raising a fraction to an exponent, apply the exponent to both the numerator and denominator: (a/b)ⁿ = aⁿ ÷ bⁿ
Example: (3/4)² = 9/16

4. Real-Life Applications of Exponents

• ✔ Computer Memory & Storage: Data is stored in powers of 2 (e.g., kilobytes, megabytes, gigabytes).
• ✔ Science & Physics: Formulas like the area of a circle, πr², use exponents.
• ✔ Finance & Interest: Compound interest follows formulas like A = P(1+r)^t.

7. Summary

• ✅ Exponents represent repeated multiplication.
• ✅ Key exponent rules:
• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^m×n
• a⁰ = 1
• ✅ Exponents are widely used in science, computing, and finance.