Study Material — 12 PYQs (2019–2019) · Concept Notes · Shortcuts
SSC CPO Surds & Indices is a frequently tested subtopic — 12 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.
- √(a × b) = √a × √b
- √(a/b) = √a / √b
- (√a)^2 = a
- √a × √a = a
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→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.
Key Points to Remember
Index law a^m × a^n = a^(m+n) works only when the base is the same
a^(m/n) = ⁿ√(a^m)—this is the bridge between indices and surds
Rationalizing means removing surds from the denominator using conjugate multiplication
√(ab) = √a × √b, but √(a+b) ≠ √a + √b—this is a critical trap
Any number to the power 0 equals 1: a^0 = 1 (except when a = 0)
Negative indices flip the fraction: a^(-n) = 1/a^n
Exam-Specific Tips
The general index law for multiplication is a^m × a^n = a^(m+n), valid for all real bases and exponents
The fractional index formula a^(m/n) = ⁿ√(a^m) allows conversion between power and root notation
A surd is an irrational root that cannot be expressed as a simple fraction (e.g., √2, √3, ∛7)
To rationalize 1/(√a + √b), multiply by (√a - √b)/(√a - √b) to eliminate surds from denominator
The law (a^m)^n = a^(mn) means nested powers multiply—critical for simplification questions
Any non-zero number raised to power 0 equals 1: a^0 = 1 (this is an axiomatic rule in SSC questions)
√a × √a = a for any positive real number a—this is used in rationalizing and simplifying
60-Second Revision — Surds & Indices
Remember: Index laws work only with the same base. Don't add exponents unless bases are identical.
Formula: a^(m/n) = ⁿ√(a^m)—use this to convert between exponent and root forms instantly.
Trap: √(a+b) ≠ √a + √b. This is a very common error. Only √(ab) = √a × √b is valid.
Rationalizing trick: For 1/(√a + √b), multiply top and bottom by (√a - √b) to use difference of squares.
Key rule: a^0 = 1 and a^(-n) = 1/a^n—these appear in almost every SSC surd question.
Practice step: Always convert surds to index form first, simplify using index laws, then convert back if needed.
Check: After simplifying, verify your answer makes sense—surds should reduce, and fractions should simplify completely.