8
CBSE Marks
★★★★★
Difficulty
8
Topics
High
Board Weight
Topics Covered
8 key topics in this chapter
Natural Numbers, Integers, Rationals
Irrational Numbers
Real Numbers & Real Number Line
Representing Reals on Number Line
Successive Magnification
Operations on Real Numbers
Laws of Exponents for Real Numbers
Rationalisation of Denominators
Study Resources
Key Formulas & Identities
| Formula / Rule | Expression |
|---|---|
| Product Rule | \(aᵐ · aⁿ = aᵐ⁺ⁿ\) |
| Quotient Rule | \(aᵐ / aⁿ = aᵐ⁻ⁿ\) |
| Power of Power | \((aᵐ)ⁿ = aᵐⁿ\) |
| Product of Powers | \((ab)ⁿ = aⁿbⁿ\) |
| Negative Exponent | \(a⁻ⁿ = 1/aⁿ\) |
| Zero Exponent | \(a⁰ = 1 (a ≠ 0)\) |
| Fractional Exponent | \(a^(1/n) = ⁿ√a\) |
| Rationalising Factor | \(1/(√a+√b) = (√a−√b)/(a−b)\) |
Important Points to Remember
Every rational number can be expressed as a terminating or repeating decimal. Every irrational number has a non-terminating, non-repeating decimal expansion.
Surds like √2, √3, √5 are irrational. They cannot be written as p/q where p, q are integers and q ≠ 0.
Rationalisation: to remove a surd from the denominator multiply numerator and denominator by the conjugate. E.g. 1/(√a+√b) × (√a−√b)/(√a−√b).
Successor magnification: to represent √n on the number line, use the right-triangle construction with legs 1 and √(n−1).