A surd is a root that never comes out exactly, such as √2 or ∛5. This course begins by defining surds, separating them from other irrational numbers like π, and naming mixed and conjugate surds. It then sets out the laws for multiplying and dividing surds, introduces rationalising factors, and teaches you to simplify surds, add like surds and find powers such as (√2)¹¹. You will multiply surds using the algebraic identities, rationalise denominators, find the value of expressions such as x³ − 2x² − 7x + 5 when x is a surd, and take the square root of a surd like 5 + 2√6.
Middle Years
Rationalization and Surds
What surds are, the laws of surds, rationalising factors, simplifying and raising surds to powers, multiplying with identities, rationalising denominators, values of expressions, and square roots of surds.
- 8 lessons
Overview
Lessons
-
What surds are
Roots that come out exactly and roots that do not, the definition of a surd, why π is irrational but not a surd, and mixed and conjugate surds.
-
The laws of surds
Six laws for multiplying and dividing surds, what the laws do not allow, and multiplying brackets that contain surds, including a conjugate pair.
-
Rationalising factors
Two surds whose product is rational are rationalising factors of each other. Find them for √a, a + √b, √a + √b and for sums of cube roots.
-
Simplifying surds and powers of surds
Write surds in simplest form, such as √72 = 6√2, change mixed surds to pure surds, add like surds, and work out powers such as (√2)¹¹ and (2√2)⁵.
-
Multiplying surds with identities
Use (a + b)², (a − b)² and (a + b)(a − b) to square binomial surds, multiply conjugate pairs and simplify general products of surds.
-
Rationalising the denominator
Remove surds from the denominator by multiplying by a rationalising factor, compare rational and surd parts, and add fractions with surd denominators.
-
Finding the value of an expression
Given a surd value of x, find the value of an expression by working out powers or by finding an equation x satisfies, and short cuts for x + 1/x.
-
The square root of a surd
Simplify roots such as √(5 + 2√6) = √3 + √2 by finding two numbers with a given sum and product, keep the answer positive, and work backwards.
Practice papers
Ten question papers that follow the lessons in order. Each has 20 questions for 50 marks (1 hour 30 minutes), with a key sheet and worked solutions.
MCQ tests
Ten tests of 25 multiple-choice questions (30 minutes each). Answer them on the page and check your score, or download the PDF with its key and solutions.
Class notes
Rationalization and Surds
Typed from the teacher's handwritten class slides, 12 pages.
Download the notes (PDF, 663.6 KB)