Learn: Surds and Simplification

Welcome!Today we'll explore surds and how to simplify them. Surds are numbers that can't be expressed as exact decimals, like the square root of 2 or 3. Let's get started!

What are Surds?A surd is an irrational number that can't be simplified to remove the square root (or cube root, etc.). For example, √2 is a surd because its decimal goes on forever without repeating. Surds are used in calculations when exact values are needed.

Simplifying SurdsTo simplify surds, break down the number inside the root into factors, one of which is a perfect square. For example, √12 can be simplified because 12 = 4 × 3, and √4 = 2. So, √12 = 2√3.

Rationalising DenominatorsSometimes surds appear in the denominator of a fraction. To simplify, we use rationalising. For example, to simplify 1/√2, multiply both numerator and denominator by √2, giving √2/2.

Operations with SurdsYou can add or subtract surds only if they have the same 'root'. For example, √3 + √3 = 2√3, but √3 + √2 cannot be simplified further. Multiplying surds is straightforward: √a × √b = √(a × b).

Exact ValuesSometimes surds are left in their exact form, especially in geometry and trigonometry. For instance, the length of the diagonal of a square with sides of length 1 is √2. Keeping values exact avoids rounding errors.

Review Time!Great work! You've learned about surds, rationalising denominators, operations with surds, and exact values. Let's test your understanding with a few questions.