Surds and Indices

This lesson covers the laws of indices and the manipulation of surds, forming the foundation of A-Level algebra as required by the Edexcel 9MA0 specification. You must be confident simplifying expressions involving fractional and negative indices, and rationalising denominators involving surds.

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Laws of Indices

The laws of indices apply to any base, provided the base is the same across terms being combined.

Law	Rule	Example
Multiplication	aᵐ × aⁿ = aᵐ⁺ⁿ	2³ × 2⁴ = 2⁷ = 128
Division	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5² = 5⁴ = 625
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(3²)⁴ = 3⁸ = 6561
Zero index	a⁰ = 1	7⁰ = 1
Negative index	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8
Fractional index	a^(1/n) = ⁿ√a	8^(1/3) = ³√8 = 2
Combined fractional	a^(m/n) = (ⁿ√a)ᵐ	27^(2/3) = (³√27)² = 9

Applying the Laws

When simplifying expressions, always convert roots to fractional powers first. For example:

Simplify √(x³) ÷ x^(1/2):

• √(x³) = x^(3/2)
• x^(3/2) ÷ x^(1/2) = x^(3/2 − 1/2) = x¹ = x

Worked Example: Simplify (2x³y²)⁴ ÷ (4x²y).

Solution:

• (2x³y²)⁴ = 2⁴ × x¹² × y⁸ = 16x¹²y⁸
• 16x¹²y⁸ ÷ 4x²y = 4x¹⁰y⁷

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Surds

A surd is an irrational number expressed as a root that cannot be simplified to a rational number. For example, √2, √3 and √5 are surds, but √4 = 2 is not.

Simplifying Surds

To simplify a surd, find the largest perfect square factor:

• √50 = √(25 × 2) = 5√2
• √72 = √(36 × 2) = 6√2
• √48 = √(16 × 3) = 4√3

Rules for Surds

Rule	Example
√a × √b = √(ab)	√3 × √5 = √15
√a ÷ √b = √(a/b)	√12 ÷ √3 = √4 = 2
(√a)² = a	(√7)² = 7
a√n + b√n = (a + b)√n	3√2 + 5√2 = 8√2

Adding and Subtracting Surds

You can only combine surds with the same radicand (the number under the root):

• 3√5 + 7√5 = 10√5
• 4√3 − √3 = 3√3
• 2√2 + 3√3 cannot be simplified further

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Expanding Brackets with Surds

Treat surds like algebraic terms. Apply the distributive law or FOIL as appropriate.

Example: Expand (2 + √3)(4 − √3).

• = 2 × 4 + 2 × (−√3) + √3 × 4 + √3 × (−√3)
• = 8 − 2√3 + 4√3 − 3
• = 5 + 2√3

Example: Expand (√5 + 1)².

• = (√5)² + 2(√5)(1) + 1²
• = 5 + 2√5 + 1
• = 6 + 2√5

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Rationalising the Denominator

A fraction with a surd in the denominator should be rationalised. The method depends on the form of the denominator.

Simple Denominator: 1/√a

Multiply numerator and denominator by √a:

• 5/√3 = (5 × √3)/(√3 × √3) = 5√3/3

Compound Denominator: 1/(a + √b)

Multiply by the conjugate (a − √b)/(a − √b):

Example: Rationalise 3/(2 + √5).

• Multiply by (2 − √5)/(2 − √5):
• Numerator: 3(2 − √5) = 6 − 3√5
• Denominator: (2 + √5)(2 − √5) = 4 − 5 = −1
• Result: (6 − 3√5)/(−1) = −6 + 3√5 = 3√5 − 6

Exam Tip: When rationalising a denominator of the form a + b√c, always use the conjugate a − b√c. The denominator becomes a² − b²c, which is rational. Show every step — examiners award method marks for clear working.

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Indices with Algebraic Expressions

At A-Level you must handle expressions such as:

Example: Solve 4ˣ = 8.

• Write both sides as powers of 2: (2²)ˣ = 2³
• 2²ˣ = 2³
• So 2x = 3, giving x = 3/2

Example: Simplify (x^(1/2) + x^(−1/2))².

• = x + 2 × x^(1/2) × x^(−1/2) + x⁻¹
• = x + 2 + 1/x

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Common Mistakes

• Confusing (aᵐ)ⁿ = aᵐⁿ with aᵐ × aⁿ = aᵐ⁺ⁿ — the first multiplies exponents, the second adds them.
• Forgetting that a^(−n) means 1/aⁿ, not −aⁿ.
• Trying to add surds with different radicands: √2 + √3 ≠ √5.
• Failing to fully simplify surds before combining them.

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Key Terms

Term	Definition
Index (exponent)	The power to which a base number is raised
Surd	An irrational root that cannot be simplified to a rational number
Rationalise	Remove the surd from the denominator of a fraction
Conjugate	The expression formed by changing the sign between two terms, e.g. the conjugate of (a + √b) is (a − √b)
Radicand	The number under the root sign

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Summary

• The laws of indices allow you to simplify expressions with the same base by adding, subtracting or multiplying exponents.
• Negative indices give reciprocals; fractional indices give roots.
• Surds are simplified by extracting the largest perfect square factor.
• Rationalise the denominator by multiplying by the appropriate form of 1 (using the conjugate where necessary).
• These skills are essential for every topic in A-Level Mathematics — you will use them repeatedly throughout the course.