AP Class 8 Mathematics
1. Rational Numbers
Properties
| Property | Addition | Multiplication |
|---|
| Closure | a+b is rational | a×b is rational |
| Commutative | a+b = b+a | a×b = b×a |
| Associative | (a+b)+c = a+(b+c) | (ab)c = a(bc) |
| Identity | 0 (a+0=a) | 1 (a×1=a) |
| Inverse | a+(−a)=0 | a×(1/a)=1 (a≠0) |
| Distributive | — | a(b+c) = ab + ac |
Representing on the Number Line
Between any TWO rational numbers, there are INFINITELY MANY rational numbers. Find by taking the MEAN: (a+b)/2.
2. Exponents and Powers
Laws of Exponents
| Law | Formula |
|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ |
| Power of Product | (ab)ᵐ = aᵐbᵐ |
| Zero Exponent | a⁰ = 1 (a≠0) |
| Negative Exponent | a⁻ⁿ = 1/aⁿ |
Large numbers: 150,000,000 km = 1.5 × 10⁸ km. Small numbers: 0.000000001 m = 1 × 10⁻⁹ m.
3. Algebraic Expressions and Identities
Key Identities
- (a+b)² = a² + 2ab + b²
- (a−b)² = a² − 2ab + b²
- (a+b)(a−b) = a² − b²
- (x+a)(x+b) = x² + (a+b)x + ab
Factorisation Methods
| Method | Example |
|---|
| Common Factor | 6x+9 = 3(2x+3) |
| Grouping | ax+bx+ay+by = x(a+b)+y(a+b) = (a+b)(x+y) |
| Difference of Squares | x²−16 = (x+4)(x−4) |
| Splitting Middle Term | x²+5x+6 = (x+2)(x+3) |
4. Mensuration
2D Shapes — Perimeter and Area
| Shape | Perimeter | Area |
|---|
| Square | 4s | s² |
| Rectangle | 2(l+w) | l×w |
| Triangle | a+b+c | ½bh |
| Circle | 2πr | πr² |
| Trapezium | Sum of all sides | ½(a+b)h |
3D Solids — Surface Area and Volume
| Solid | TSA | Volume |
|---|
| Cube | 6s² | s³ |
| Cuboid | 2(lb+bh+hl) | lbh |
| Cylinder | 2πr(r+h) | πr²h |
5. Data Handling
Organising Data
Frequency distribution tables. Grouped data. Histograms (continuous data — bars TOUCH). Pie charts (360° = 100%).
Probability
P(E) = Number of favourable outcomes / Total outcomes. Coins. Dice. Cards.
Common Mistakes to Avoid
- Forgetting the negative sign when simplifying: (−3)² = +9, but −3² = −9. 'Parentheses MATTER.'
- In factorisation, NOT checking by expanding: 'ALWAYS expand your factors to verify — they must equal the original expression.'
- TSA of cylinder includes TOP and BOTTOM circles: 'TSA = 2πr² + 2πrh = 2πr(r+h). Don't forget the circles!'
