📚 Class 7 – Mathematics
Chapter: Exponents & Powers
Goal: Learn how to read, write, and work with powers (exponents) using simple rules and everyday language.
1️⃣ What Is a Power (Exponent)?
| Symbol | Meaning | Everyday Analogy |
|---|---|---|
aⁿ |
a multiplied by itself n times | “Three apples on each of 4 plates” → 4 × 4 × 4 = 4³ |
- Base = the number being multiplied (here
a). - Exponent = how many times the base repeats (here
n).
Tiny Example
3²means3 × 3 = 9.
(Three multiplied by itself twice.)
2️⃣ Basic Rules – Short Sentences & Tiny Examples
2.1. Multiplying Powers with the Same Base
aᵐ × aⁿ = a^(m+n)
Add the exponents because you are putting the same base side by side.
| Example | Calculation | Result |
|---|---|---|
2² × 2³ |
2 × 2 × 2 × 2 × 2 × 2 |
2⁵ = 32 |
2.2. Dividing Powers with the Same Base
aᵐ ÷ aⁿ = a^(m−n)
Subtract the exponents because you are taking one product away from another.
| Example | Calculation | Result |
|---|---|---|
5⁴ ÷ 5² |
5 × 5 × 5 × 5 ÷ 5 × 5 |
5² = 25 |
2.3. Power of a Product
(ab)ⁿ = aⁿ × bⁿ
Raise each factor separately then multiply.
| Example | Calculation | Result |
|---|---|---|
(2 × 3)² |
2² × 3² = 4 × 9 |
36 |
2.4. Power of a Power
(aᵐ)ⁿ = a^(m×n)
Multiply the two exponents.
| Example | Calculation | Result |
|---|---|---|
(3²)³ |
3^(2×3) = 3⁶ |
729 |
2.5. Difference of Squares
a² – b² = (a – b)(a + b)
Think of it as “square minus square” → “difference × sum”.
| Example | Calculation | Result |
|---|---|---|
5² – 2² |
(5 – 2)(5 + 2) = 3 × 7 |
21 |
2.6. Sum or Difference of Cubes
(Only for cubes, not needed in every test but handy to know.)
| Formula | Tiny Example |
|---|---|
a³ + b³ = (a + b)(a² – ab + b²) |
4³ + 1³ = (4 + 1)(16 – 4 + 1) = 5 × 13 = 65 |
a³ – b³ = (a – b)(a² + ab + b²) |
`3³ – 2³ = (3 |