Rational Numbers, Exponents, Squares and Roots: Class 8 Study Guide
Number properties make calculation faster and show how numbers are related.
Rational numbers are closed under addition, subtraction and multiplication (the result is always rational) and under division except by zero. They obey the commutative, associative and distributive properties. The additive identity is 0, the multiplicative identity is 1, and every non-zero rational number a/b has a reciprocal b/a.
Exponents show repeated multiplication. The laws, for non-zero a: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a⁰ = 1; and a^(−n) = 1/a^n. A very large or small number can be written in standard form, such as 3.2 × 10⁶ = 3 200 000 or 4.5 × 10⁻³ = 0.0045.
Laws of exponents
| Law | Example |
|---|---|
| a^m × a^n = a^(m+n) | 2³ × 2⁴ = 2⁷ = 128 |
| a^m ÷ a^n = a^(m−n) | 5⁶ ÷ 5⁴ = 5² = 25 |
| (a^m)^n = a^(mn) | (3²)³ = 3⁶ = 729 |
| a⁰ = 1 | 7⁰ = 1 |
| a^(−n) = 1/aⁿ | 2⁻³ = 1/8 |
Squares and square roots
A perfect square is the square of a whole number. 1, 4, 9, 16, 25, 36… A number ending in 2, 3, 7 or 8 is never a perfect square. To find a square root by prime factorisation: 144 = 2⁴ × 3², so √144 = 2² × 3 = 12. For larger numbers, the long division method gives √ to any accuracy. The square of a number with n digits has 2n or 2n − 1 digits.
144 = 2 × 2 × 2 × 2 × 3 × 3.
(2 × 2)(2 × 2)(3 × 3).
2 × 2 × 3.
12.
Cubes and cube roots
A perfect cube is the cube of a whole number: 1, 8, 27, 64, 125… To find ∛ by factorisation, group the prime factors in threes: 216 = 2³ × 3³, so ∛216 = 2 × 3 = 6. Patterns help in estimating: ∛1000 = 10 and ∛8000 = 20.
Worked examples and common mistakes
Example: Simplify (2³ × 2⁵) ÷ 2⁴ = 2⁸ ÷ 2⁴ = 2⁴ = 16. Common mistakes: adding bases (2³ × 3³ ≠ 5⁶), treating (−3)² as −9 and believing that √(a + b) = √a + √b. Check with small numbers.
Check your understanding: model answers
1. 3³ = 27. 2. 15. 3. 7.2 × 10⁻³.
✨ Key Takeaways
- Rational numbers are closed under +, − and ×.
- Know the laws of exponents.
- Use prime factors for roots.
- Standard form handles very large and small numbers.
💬 Questions for Discussion & Reflection
- Simplify 3⁵ × 3² ÷ 3⁴.
- Find √225.
- Write 0.0072 in standard form.