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Surds & Indices

CRPF Constable · Quantitative Aptitude

Also prepare for:BSF Const CISF Const ITBP Const UP Const

← BODMAS

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Core Concept

Read this first — the foundation of the topic

→Index Laws

- a^m × a^n = a^(m+n) - a^m ÷ a^n = a^(m-n) - (a^m)^n = a^(mn) - a^0 = 1 (for any a ≠ 0) - a^(-n) = 1/a^n 2

🔑Surd Rules

- √(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

1

Step 1

Convert to surd form 16^(3/4) = ⁴√(16³) = (⁴√16)³ = 2³ = 8

2

Step 2

Simplify the denominator 8^(-2/3) = 1/(8^(2/3)) = 1/(³√(8²)) = 1/(³√64) = 1/4

3

Step 3

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

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 Facts

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

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.

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