- β(a Γ b) = βa Γ βb
- β(a/b) = βa / βb
- (βa)^2 = a
- βa Γ βa = a
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βRationalizing Surds
Remove surds from the denominator by multiplying numerator and denominator by the conjugate.
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Formula Block
Memorise β at least one formula appears in every paper
a^(m/n) = βΏβ(a^m) β This connects indices and surds. For example, 8^(2/3) = Β³β(8Β²) = Β³β64 = 4
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Exam Patterns
What examiners ask β 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/(βa + βb), multiply by (βa - βb)/(βa - βb). This uses the difference of squares formula: (x+y)(x-y) = xΒ² - yΒ². 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) = β΄β(16Β³) = (β΄β16)Β³ = 2Β³ = 8
Divide
8 Γ· (1/4) = 8 Γ 4 = 32
COMMON MISTAKE:
Students often forget that βa Γ βb β β(a+b). The correct rule is βa Γ βb = β(ab). Also, many forget that a^(1/2) = βa, not a/2.