Simplification of Surds: Formulas, Rules and Examples
Simplification of Surds means reducing radical expressions to their simplest form by extracting perfect powers, combining like surds and rationalising denominators. This page covers square-root and higher-root rules, surd formulas, factorisation methods, common surd questions and quick calculation techniques with examples.
What is Simplification of Surds?
For example, √72 = √(36 × 2) = 6√2. The factor 36 is a perfect square, so it comes outside the square root. A simplified surd has no perfect square factor inside a square root, no perfect cube factor inside a cube root, and so on. The value of the expression remains unchanged.
Simplification of Surds Formula & Tricks
Important Formulas
Square roots can be multiplied by multiplying their radicands.
A square root of a quotient can be separated into the quotient of the square roots.
A perfect square factor comes outside the square root. If a is positive, this becomes a√b.
A perfect nth power can be taken outside an nth root.
Multiply the numerator and denominator by the conjugate, using (x + y)(x − y) = x² − y².
Quick Tricks
Factor the radicand using the largest possible perfect square. This gives the shortest square-root form.
Do not add or subtract radicals until each one has been reduced. Then combine only terms with the same radical part.
For a denominator such as √a + √b, multiply by √a − √b. For √a − √b, use √a + √b.
A radical such as √49 or ∛64 is rational and should be evaluated completely rather than left as a surd.
Simplification of Surds Concepts
Reducing Square-Root Surds by Factorisation
Use √(a²b) = a√b for a positive integer a. Continue factoring until the remaining radicand has no perfect-square factor greater than 1. For example, √48 = √(16 × 3) = 4√3.
Simplifying Cube Roots and Higher Surds
Apply ⁿ√(aⁿb) = aⁿ√b for non-negative a. For example, ∛54 = ∛(27 × 2) = 3∛2. Similarly, ⁴√(80) = ⁴√(16 × 5) = 2⁴√5.
Addition and Subtraction of Like Surds
First simplify every surd. Terms such as 3√5 and −2√5 are like surds, but √2 and √3 are unlike surds. Thus, 4√7 − √7 = 3√7, whereas √2 + √3 cannot be reduced by addition.
Multiplication and Division of Surds
For multiplication, √a × √b = √(ab). For division, √a/√b = √(a/b) when the expressions are defined. Simplify the resulting radical after performing the operation.
Rationalising a Surd Denominator
For a denominator √a, multiply numerator and denominator by √a. For a binomial denominator, multiply by its conjugate. For example, 3/√5 = (3√5)/5, and 2/(√3 + 1) = 2(√3 − 1)/(3 − 1) = √3 − 1.
Simplification of Surds Video Lessons
Watch short topic-wise lessons for quick revision.
Surds in Simplification: Rationalisation
Learn how to simplify surds and rationalise denominators using standard methods, with clear steps for solving surd-based simplification problems accurately.
Practice Simplification of Surds Questions
Practise published questions related to this topic.
Simplification of Surds Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Simplification of Surds questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Simplification of Surds: Quick Revision
Use these rules to reduce and operate on radical expressions.
- Extract perfect square factors from square roots and perfect cube factors from cube roots.
- √(a²b) = |a|√b; when a is positive, write a√b.
- Simplify every surd before adding or subtracting terms.
- Only like surds can be combined.
- √a × √b = √(ab) for non-negative a and b.
- √(a/b) = √a/√b for a ≥ 0 and b > 0.
- Use the conjugate to rationalise a denominator containing two terms.
- Do not split √(a + b) as √a + √b; this identity is generally false.
Simplification of Surds FAQs
How do you simplify √72?
Factor 72 as 36 × 2: √72 = √(36 × 2) = 6√2.
Can √2 and √8 be added directly?
No. First simplify √8 = 2√2. Therefore, √2 + √8 = √2 + 2√2 = 3√2.
What is the difference between √(a + b) and √a + √b?
They are generally not equal. For example, √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7.
How do you simplify √50 × √8?
√50 × √8 = √400 = 20. Equivalently, 5√2 × 2√2 = 10 × 2 = 20.
How do you rationalise 1/√3?
Multiply the numerator and denominator by √3: 1/√3 = √3/3.
How do you rationalise 1/(√5 − √2)?
Multiply by the conjugate: 1/(√5 − √2) = (√5 + √2)/(5 − 2) = (√5 + √2)/3.