Surds

1. Overview

Surds are irrational numbers expressed using a radical symbol (√), most commonly a square root. Because their decimal representations are non-terminating and non-repeating, surds provide exact values in calculations, avoiding rounding errors. This is especially important in fields requiring precision. This revision note covers simplifying, manipulating, and rationalising surds, as required for the IGCSE Cambridge Mathematics (0580) Extended syllabus.

Key Definitions

• Surd: An irrational number that is expressed using a radical sign (usually a square root).
• Rational Number: A number that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers.
• Irrational Number: A number that cannot be written as a simple fraction; its decimal expansion goes on forever without repeating.
• Rationalising the Denominator: The process of modifying a fraction so that the denominator contains only rational numbers (no surds).
• Conjugate pair: Expressions like $(a + \sqrt{b})$ and $(a - \sqrt{b})$ which, when multiplied, result in a rational number.

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Core Content

Note: This topic is entirely within the Extended (Supplement) curriculum. There are no Core-only objectives for Surds.

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Extended Content (Extended Curriculum Only)

A. Simplifying Surds

To simplify a surd, find the largest perfect square that is a factor of the number under the square root. This allows you to express the surd in its simplest form. Rule: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

Worked example 1 — Simplifying a surd

Question: Simplify $\sqrt{108}$

1. Identify the largest perfect square factor of 108: $36 \times 3 = 108$ $$\sqrt{108} = \sqrt{36 \times 3}$$ Reason: Express 108 as a product of its largest perfect square factor and another factor.
2. Apply the rule $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$: $$\sqrt{36} \times \sqrt{3}$$ Reason: Separate the square root of the product into the product of square roots.
3. Calculate the square root of the perfect square: $$6 \times \sqrt{3}$$ Reason: $\sqrt{36} = 6$
4. Write in simplest form: $$\mathbf{6\sqrt{3}}$$

B. Adding and Subtracting Surds

You can only add or subtract "like" surds (surds with the same number under the square root), similar to collecting like terms in algebra. Simplify each surd individually before attempting to add or subtract.

Worked example 2 — Adding surds

Question: Simplify $3\sqrt{8} + \sqrt{50} - \sqrt{2}$

1. Simplify $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$ Reason: Find the largest perfect square factor of 8.
2. Simplify $\sqrt{50}$: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$ Reason: Find the largest perfect square factor of 50.
3. Rewrite the original expression with simplified surds: $$3(2\sqrt{2}) + 5\sqrt{2} - \sqrt{2}$$ Reason: Substitute the simplified forms of $\sqrt{8}$ and $\sqrt{50}$.
4. Multiply: $$6\sqrt{2} + 5\sqrt{2} - \sqrt{2}$$ Reason: Multiply the coefficient.
5. Combine the terms: $$(6 + 5 - 1)\sqrt{2}$$ Reason: Add and subtract the coefficients of the like surds.
6. Simplify: $$\mathbf{10\sqrt{2}}$$

C. Multiplying Surds

When multiplying surds, multiply the numbers outside the roots together and the numbers inside the roots together. Rule: $x\sqrt{a} \times y\sqrt{b} = (xy)\sqrt{ab}$

Worked example 3 — Multiplying surds

Question: Expand and simplify $(2\sqrt{3} - 1)(\sqrt{3} + 4)$

1. Expand the brackets using FOIL (First, Outer, Inner, Last): $$(2\sqrt{3} \times \sqrt{3}) + (2\sqrt{3} \times 4) + (-1 \times \sqrt{3}) + (-1 \times 4)$$ Reason: Apply the distributive property to expand the product of the two binomials.
2. Multiply each term: $$(2 \times 3) + 8\sqrt{3} - \sqrt{3} - 4$$ Reason: Simplify each multiplication, noting that $\sqrt{3} \times \sqrt{3} = 3$.
3. Simplify: $$6 + 8\sqrt{3} - \sqrt{3} - 4$$ Reason: Calculate 2 x 3.
4. Combine like terms: $$(6 - 4) + (8\sqrt{3} - \sqrt{3})$$ Reason: Group the constant terms and the terms with $\sqrt{3}$.
5. Simplify: $$2 + 7\sqrt{3}$$ Reason: Combine the constant terms and the coefficients of $\sqrt{3}$. Answer: $2 + 7\sqrt{3}$

D. Rationalising the Denominator

Case 1: Denominator is a single surd ($\frac{k}{\sqrt{a}}$) Multiply the numerator and denominator by $\sqrt{a}$.

Worked example 4 — Rationalising a single-term surd denominator

Question: Rationalise $\frac{6}{\sqrt{3}}$

1. Multiply numerator and denominator by $\sqrt{3}$: $$\frac{6 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}$$ Reason: Multiply by $\frac{\sqrt{3}}{\sqrt{3}}$, which is equal to 1, so the value of the fraction doesn't change.
2. Simplify the denominator ($\sqrt{3} \times \sqrt{3} = 3$): $$\frac{6\sqrt{3}}{3}$$ Reason: Simplify the denominator.
3. Simplify the fraction: $$\mathbf{2\sqrt{3}}$$ Reason: Divide 6 by 3.

Case 2: Denominator is a binomial ($a \pm \sqrt{b}$) Multiply the numerator and denominator by the conjugate (change the sign).

Worked example 5 — Rationalising a binomial surd denominator

Question: Rationalise $\frac{2}{5 + \sqrt{3}}$

1. Identify the conjugate of the denominator: $5 - \sqrt{3}$ Reason: The conjugate is formed by changing the sign between the terms.
2. Multiply numerator and denominator by $(5 - \sqrt{3})$: $$\frac{2(5 - \sqrt{3})}{(5 + \sqrt{3})(5 - \sqrt{3})}$$ Reason: Multiply by $\frac{5 - \sqrt{3}}{5 - \sqrt{3}}$, which is equal to 1.
3. Expand the numerator and denominator: $$\frac{10 - 2\sqrt{3}}{25 - 5\sqrt{3} + 5\sqrt{3} - 3}$$ Reason: Expand both the numerator and the denominator.
4. Simplify the denominator: $$\frac{10 - 2\sqrt{3}}{22}$$ Reason: The middle terms cancel out, and $\sqrt{3} \times \sqrt{3} = 3$.
5. Simplify the fraction (if possible): $$\frac{2(5 - \sqrt{3})}{2(11)}$$ Reason: Factor out a 2 from the numerator and denominator.
6. Cancel the common factor: $$\mathbf{\frac{5 - \sqrt{3}}{11}}$$ Reason: Simplify the fraction.

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Key Equations

These rules are not provided on the formula sheet and must be memorised:

Rule	Equation
Multiplication	$\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
Division	$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
Squaring a Root	$(\sqrt{a})^2 = a$
Conjugate Expansion	$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$

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Common Mistakes to Avoid

• ❌ Wrong: $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$ (Incorrectly distributing the square root)
• ✅ Right: $\sqrt{a + b}$ cannot be simplified directly unless $a + b$ is a perfect square. For example, $\sqrt{9 + 16} = \sqrt{25} = 5$.
• ❌ Wrong: Expanding $(a + \sqrt{b})^2$ as $a^2 + b$ (Forgetting the middle term)
• ✅ Right: $(a + \sqrt{b})^2 = (a + \sqrt{b})(a + \sqrt{b}) = a^2 + 2a\sqrt{b} + b$
• ❌ Wrong: Not simplifying after rationalising, e.g., leaving the answer as $\frac{4 + 2\sqrt{3}}{2}$
• ✅ Right: Always simplify after rationalising: $\frac{4 + 2\sqrt{3}}{2} = \frac{2(2 + \sqrt{3})}{2} = 2 + \sqrt{3}$
• ❌ Wrong: Incorrectly identifying the conjugate. For example, thinking the conjugate of $2 + \sqrt{5}$ is $-2 - \sqrt{5}$.
• ✅ Right: The conjugate of $2 + \sqrt{5}$ is $2 - \sqrt{5}$. Only the sign between the rational and irrational parts changes.

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Exam Tips

• Show Every Step: Especially in "Show that" questions, meticulously display each simplification step. Omitting steps can lead to significant mark deductions.
• Check for Square Factors: After obtaining a surd in your answer (e.g., $\sqrt{75}$), always verify if it can be further simplified by identifying square factors ($\sqrt{25 \times 3} = 5\sqrt{3}$).
• Command Words:
  • "Give your answer in the form $a\sqrt{b}$": This explicitly instructs you to simplify the surd to its simplest form, where $a$ and $b$ are integers and $b$ has no square factors.
  • "Rationalise the denominator": This command requires you to eliminate any surds from the denominator of a fraction. Failure to do so will result in lost marks.
• Calculator Tip: Utilize your calculator to verify your surd simplifications. However, remember to demonstrate the manual steps in your working to receive full credit.
• Common Values: Familiarise yourself with perfect squares up to $15^2$ (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225) to expedite factor identification during simplification.