Real Numbers and Surds
Real Numbers and Surds tests whether you can move between fractions, radicals and exponent form without slipping on signs or indices. Expect simplification, comparison and rationalisation.
Core rules
- Every surd is irrational, but not every irrational is a surd: π and e are irrational without being square-root expressions.
- For positive a, b: √(ab) = √a√b and √(a/b) = √a/√b, but √(a+b) ≠ √a + √b.
- To rationalise 1/(√a + √b), multiply numerator and denominator by the conjugate √a − √b.
- a^(m/n) = (n√a)^m, and a^m × a^n = a^(m+n). Keep the base positive when the index is even.
Worked example
Simplify √50 + √18 − √8. Write each surd in simplest form: √50 = √(25×2) = 5√2; √18 = 3√2; √8 = 2√2. Then 5√2 + 3√2 − 2√2 = 6√2. The terms combine only because they share the same irrational part.
Traps
A common error is writing √9+16 = 3+4; actually √25 = 5. In rationalisation, forgetting to multiply the numerator as well changes the value. When comparing 2√3 and 3√2, square both: 12 < 18, so 2√3 < 3√2. Also check whether the question wants a simplified surd or a decimal approximation - they are not interchangeable.