Surds: Formulas, Rules and Simplification Methods

Surds are irrational roots written in exact form, such as √2, √3 and ³√5. This page explains surds basics, simplification rules, operations on surds, conjugates, rationalisation and frequently used formulas. Worked examples show how to reduce surds to their simplest forms and solve common quantitative aptitude questions.

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Overview Surds Formula & Tricks Surds Concepts Surds Video Lessons Practice Surds Questions Surds Quick Quiz Surds Revision Notes Surds FAQs

What Are Surds?

A surd is an irrational root that cannot be expressed as a rational number. Examples include √2, √5 and ³√7, while √4 = 2 is not a surd.

A root is in simplest surd form when no perfect square factor remains inside a square root, no perfect cube factor remains inside a cube root, and so on. For example, √12 = √(4 × 3) = 2√3. The numbers outside and inside the radical are normally taken to be rational numbers.

Surds Formula & Tricks

Important Formulas

Product of square roots

√a × √b = √(ab), for a ≥ 0 and b ≥ 0

Multiply the radicands and then simplify the resulting surd.

Quotient of square roots

√a ÷ √b = √(a/b), for a ≥ 0 and b > 0

Divide the radicands when the denominator is non-zero.

Square of a binomial containing surds

(√a ± √b)² = a + b ± 2√(ab)

Use the identity (x ± y)² = x² + y² ± 2xy.

Difference of squares

(√a + √b)(√a − √b) = a − b

Conjugate surds remove the radical terms through the identity (x + y)(x − y) = x² − y².

Rationalisation of a single square root

1/√a = √a/a, for a > 0

Multiply the numerator and denominator by √a so that the denominator becomes rational.

Rationalisation using a conjugate

1/(√a + √b) = (√a − √b)/(a − b), for a ≠ b

Multiply by the conjugate √a − √b to remove the surd from the denominator.

Quick Tricks

Extract the largest perfect-square factor

For √N, factor N using the largest perfect square. Take its square root outside the radical.

Example: √72 = √(36 × 2) = 6√2.

Combine only like surds

Surds can be added or subtracted only after their radical parts are identical. First simplify every surd.

Example: 2√12 + √27 = 2(2√3) + 3√3 = 7√3.

Use the conjugate for rationalisation

The conjugate changes the sign between two terms and converts the denominator into a difference of squares.

Example: 1/(√5 + 2) = (√5 − 2)/(5 − 4) = √5 − 2.

Compare positive square roots by squaring

For non-negative quantities, compare their squares instead of approximating the roots.

Example: To compare √7 and 2√2, compare 7 and 8. Therefore, √7 < 2√2.

Surds Concepts

Simplifying Square-Root Surds

To simplify √N, separate N into a perfect-square factor and the remaining factor.

Use √(ab) = √a × √b. If a is a perfect square, replace √a with its rational square root. Continue until the radicand has no perfect-square factor greater than 1.

Example: √180 = √(36 × 5) = 6√5.

Addition and Subtraction of Surds

Only like surds can be added or subtracted; like surds have the same simplified radical part.

First simplify each term and then combine the rational coefficients. Unlike terms such as √2 and √3 cannot be combined into one surd.

Example: 3√8 − √18 = 3(2√2) − 3√2 = 3√2.

Multiplication and Division of Surds

Multiply or divide the coefficients and the radical parts separately, then simplify the result.

For example, (p√a)(q√b) = pq√(ab). For division, (p√a)/(q√b) can be simplified using √a/√b = √(a/b), provided q and b are non-zero where required.

Example: (2√3)(5√6) = 10√18 = 30√2.

Conjugate Surds and Rationalisation

The conjugate of a + √b is a − √b, and the conjugate of a − √b is a + √b.

Multiplying a binomial surd by its conjugate removes the middle terms: (a + √b)(a − √b) = a² − b. Rationalisation means removing surds from a denominator by multiplying by a suitable surd or conjugate.

Example: 1/(3 + √2) = (3 − √2)/(9 − 2) = (3 − √2)/7.

Comparing Surds

Non-negative surds can be compared by squaring both sides, because squaring preserves the order of non-negative numbers.

Before squaring, ensure both expressions are non-negative. For expressions with rational and irrational parts, rationalisation or a suitable algebraic comparison may be needed.

Example: Compare √11 and 10/3. Their squares are 11 and 100/9. Since 99/9 < 100/9, √11 < 10/3.