Surds are a crucial component of the IGCSE maths curriculum, offering students the opportunity to explore the intricacies of irrational numbers. A surd is an irrational number that can be expressed in the form of a square root, cube root, or any higher root that cannot be simplified to a rational number. Understanding surds is essential for simplifying expressions and solving equations that involve roots.

What are Surds?

• Definition: A surd is an irrational number that can be expressed as the root of a positive integer. Unlike rational numbers, surds cannot be expressed as a simple fraction, making their exact value impossible to determine.
• Importance: Surds maintain the precision of calculations in roots, as they represent the exact values.

Simplifying Surds

Simplifying surds involves reducing the expression to its simplest radical form without changing its value.

• Basic Principle: The simplification process relies on the property that $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
• Example: Simplifying $\sqrt{20}$ involves recognising that 20 can be expressed as $4 \times 5$. Since $\sqrt{4}$ is $2$, we can simplify $\sqrt{20}$ to $2\sqrt{5}$.

Image courtesy of Online Math Learning

Example:

• Question: Simplify $\sqrt{72}$.
• Solution:
  • Recognise that 72 can be expressed as $36 \times 2$.
  • Since $\sqrt{36}$ is $6$, $\sqrt{72}$ simplifies to $6\sqrt{2}$.

Operations with Surds

Performing operations with surds requires understanding their algebraic properties.

Addition and Subtraction

• Key Concept: Only like surds (surds with the same radicand) can be directly added or subtracted.
• Example: $\sqrt{2} + 3\sqrt{2} = 4\sqrt{2}),$ but $\sqrt{2} + \sqrt{3}$ cannot be simplified further.

Multiplication and Division

• Multiplication: Multiply the coefficients and the surds separately.
  • Example: $2\sqrt{3} \times 3\sqrt{2} = 6\sqrt{6}$
• Division: Divide the coefficients and rationalize the denominator if necessary.
  • Example: $\frac{\sqrt{10}}{\sqrt{5}} = \sqrt{2}$

Rationalizing the Denominator

This involves altering the form of a fraction so that the denominator is rational.

• Technique: Multiply the numerator and the denominator by the conjugate of the denominator.
• Example: To rationalize $\frac{1}{\sqrt{3}}$, multiply by $\frac{\sqrt{3}}{\sqrt{3}}$ to get $\frac{\sqrt{3}}{3}$.

Worked Examples

Example 1: Simplify $\sqrt{200} - \sqrt{32}$.

Solution:

1. Simplify $\sqrt{200}$:

• Prime factorization of 200: $200 = 2^3 \times 5^2$
• $\sqrt{200} = \sqrt{2^3 \times 5^2} = \sqrt{2^2 \times 5^2 \times 2} = \sqrt{4 \times 25 \times 2} = \sqrt{100 \times 2}$
• Thus, $\sqrt{200} = 10\sqrt{2}$

2. Simplify $\sqrt{32}$:

• Prime factorization of 32: $32 = 2^5$
• $\sqrt{32} = \sqrt{2^5} = \sqrt{2^4 \times 2} = \sqrt{16 \times 2}$
• Thus, $\sqrt{32} = 4\sqrt{2}$

3. Combine the expressions:

• $\sqrt{200} - \sqrt{32} = 10\sqrt{2} - 4\sqrt{2}$
• Therefore, $\sqrt{200} - \sqrt{32} = 6\sqrt{2}$.

Example 2: Rationalize and simplify $\frac{\sqrt{5}}{\sqrt{10}}$.

Solution:

1. Rationalize the denominator:

• To rationalize $\frac{\sqrt{5}}{\sqrt{10}}$, multiply both numerator and denominator by (\sqrt{10}) to get rid of the surd in the denominator.
• $\frac{\sqrt{5}}{\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{\sqrt{5} \times \sqrt{10}}{10}$

2. Simplify the expression:

• $\frac{\sqrt{5} \times \sqrt{10}}{10} = \frac{\sqrt{50}}{10}$
• Prime factorization of 50: $50 = 2 \times 5^2$
• $\frac{\sqrt{50}}{10} = \frac{\sqrt{2 \times 5^2}}{10} = \frac{5\sqrt{2}}{10}$
• Therefore, $\frac{\sqrt{5}}{\sqrt{10}} = \frac{\sqrt{2}}{2}$.