Surds and Indices

5 min

Master fundamental rules of powers, radicals, and rationalisation for competitive exams.

The Golden Rule

To simplify expressions in surds and indices, systematically apply the laws of powers, combine like radicals, and eliminate roots from denominators using rationalisation.

Why It Matters

Direct simplification questions appear in SSC, Railway, and Defence exams.
Crucial for solving complex surd equations and number system problems.
Speeds up calculation in arithmetic and algebra simplification sections.

Memory Hooks

Same base in multiplication means add the powers: a^m × a^n = a^(m+n).
Power of a power means multiply the exponents: (a^m)^n = a^(mn).
Conjugate pairs destroy roots in the denominator: (a+√b)(a-√b) = a² - b.

Laws of Indices & Surds

Operation	Formula / Rule	Example
Product Law	a^m × a^n = a^(m+n)	2³ × 2² = 2⁵
Quotient Law	a^m / a^n = a^(m-n)	5⁵ / 5³ = 5²
Negative Power	a^(-n) = 1 / a^n	3⁻² = 1/9
Surd Multiplication	ⁿ√a × ⁿ√b = ⁿ√(ab)	√2 × √8 = √16 = 4

Visual Insights

a^(1/n)

(Fractional Index Root)

Fractional Index Root

8^(1/3) equals cube root of 8 which is 2.

Rationalise

(Clear Denominator Roots)

Clear Denominator Roots

Multiply numerator and denominator by the conjugate.

Comparison

(Equalise Indices)

Equalise Indices

Compare ³√2 and √3 by taking LCM of indices 3 and 2.

Infinite Surd

(Nested Radical Trick)

Nested Radical Trick

√(12 + √(12 + ...)) splits into 4 × 3, giving answer 4.

Exam Traps & Pitfalls

(a + b)² = a² + b²(a + b)² = a² + 2ab + b²

Students often miss the middle 2ab term when expanding squared binomials.

√a + √b = √(a + b)√a + √b cannot be combined directly

Square roots do not add up under a single radical sign.

-2² = 4-2² = -4

Without parentheses, the square applies only to 2, not the negative sign.

Exam Practice Quiz

Q1.If 2^(x+3) = 32, what is the value of x?

Q2.Simplify: 1 / (√7 - √6)

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