Radical Simplification
This topic covers de-nesting compound surds and comparing surds without a calculator. A compound surd √(a ± 2√b) can often be written as √x ± √y, where x + y = a and xy = b — find two numbers that add to a and multiply to b. For example √(7 + 2√12): need x + y = 7, xy = 12, so x = 4, y = 3, giving √4 + √3 = 2 + √3. If the surd reads √(a ± √b) with no 2 in front, first rewrite it as √(a ± 2√(b/4)) to match the template. To compare surds of different orders, raise both to the LCM of their root indices: ∛2 versus √3 becomes, raising to the 6th power, 2^2 = 4 versus 3^3 = 27, so √3 is larger. To compare same-order surds, just compare the radicands. These two skills — de-nesting and ranking — are exactly what CAT tests when surds show up.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Laws of indices
| Product of same base | a^m × a^n = a^(m+n) |
|---|---|
| Quotient of same base | a^m ÷ a^n = a^(m−n) |
| Power of a power | (a^m)^n = a^(m×n) |
| Power of a product / quotient | (ab)^n = a^n b^n; (a/b)^n = a^n / b^n |
| Zero and negative index | a^0 = 1; a^(−n) = 1 / a^n |
Surds & rationalization
| Fractional index = root | a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m) |
|---|---|
| Product / quotient of roots | ⁿ√a × ⁿ√b = ⁿ√(ab); ⁿ√a ÷ ⁿ√b = ⁿ√(a/b) |
| Conjugate of (√a + √b) | √a − √b (their product = a − b) |
| Rationalize 1/(√a + √b) | (√a − √b) / (a − b) |
| Compound surd √(a ± 2√b) | √x ± √y where x + y = a, xy = b |