- √(a × b) = √a × √b
- √(a/b) = √a / √b
- (√a)^2 = a
- √a × √a = a
3
→Rationalizing Surds
Remove surds from the denominator by multiplying numerator and denominator by the conjugate.
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Formula Block
Memorise — at least one formula appears in every paper
a^(m/n) = ⁿ√(a^m) — This connects indices and surds. For example, 8^(2/3) = ³√(8²) = ³√64 = 4
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Exam Patterns
What examiners ask — read before attempting PYQs
SSC CGL usually asks:
- Simplify expressions using index laws
- Rationalize denominators containing surds
- Convert between index and surd notation
- Find unknown exponents in equations
- Compare surd values
SHORTCUT/TRICK:
When rationalizing 1/(√a + √b), multiply by (√a - √b)/(√a - √b). This uses the difference of squares formula: (x+y)(x-y) = x² - y². The denominator becomes a - b instantly, removing all surds.
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Worked Example
Solve this step-by-step before moving on
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Step 1
Convert to surd form
16^(3/4) = ⁴√(16³) = (⁴√16)³ = 2³ = 8
Divide
8 ÷ (1/4) = 8 × 4 = 32
COMMON MISTAKE:
Students often forget that √a × √b ≠ √(a+b). The correct rule is √a × √b = √(ab). Also, many forget that a^(1/2) = √a, not a/2.