Welcome to the course material on Surds in Further Mathematics. Surds are an essential component of mathematical expressions, commonly encountered in various mathematical problems. The concept of surds involves irrational numbers expressed in the form √a, where a is a positive integer that is not a perfect square. This topic aims to deepen your understanding of surds and equip you with the necessary skills to manipulate them effectively in mathematical operations.

Understanding the concept of surds is fundamental to mastering this topic. Surds often appear in equations and expressions, requiring a solid grasp of their properties and operations. Surds are typically simplified by removing any perfect square factors under the root sign, leaving the expression in its simplest form.

Performing the four basic operations on surds – addition, subtraction, multiplication, and division – is a key aspect of this topic. Addition and subtraction of surds involve combining like terms by ensuring that the root values are the same before performing the operation. Multiplication and division of surds require careful manipulation to simplify the expressions and obtain the final result in the most simplified form.

One important technique in dealing with surds is rationalizing the denominator. When surds appear in the denominator of a fraction, rationalizing involves removing the radical from the denominator by multiplying both the numerator and denominator by an appropriate expression that eliminates the radical. This process results in a rationalized form of the expression, making it easier to work with and interpret.

Moreover, the application of surds extends beyond mathematical calculations to real-life situations. Surds are commonly used in fields such as engineering, physics, and finance to represent quantities that involve square roots of numbers. Understanding and applying surds in practical scenarios enhance problem-solving skills and equip you with the necessary tools to tackle complex mathematical problems.

As we delve deeper into the realm of surds, we will explore set theory concepts that complement the understanding and manipulation of surds. The notion of sets defined by properties, set notations, Venn diagrams, and the use of sets in solving problems will further enrich your grasp of mathematical concepts and their applications.

In conclusion, this course material on Surds aims to enhance your proficiency in handling irrational numbers, performing operations on surds, rationalizing expressions, and applying these skills to real-world scenarios. By the end of this course, you will be well-equipped to tackle challenging mathematical problems involving surds with confidence and precision.


  1. Recognize and apply surds in real-life situations
  2. Perform addition, subtraction, multiplication, and division of surds
  3. Apply surds in problem-solving
  4. Understand the concept of surds
  5. Simplify surds using rationalizing the denominator method

Lesson Note

In mathematics, a surd is an expression containing a root symbol (√) which cannot be simplified to remove the root. Surds are often used in various areas of mathematics because they allow for the exact representation of irrational numbers.

Lesson Evaluation

Congratulations on completing the lesson on Surds. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.

You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.

Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.

  1. Perform the following operation on the surds: √18 + √32 A. 5√2 B. 4√2 C. 7√2 D. 6√2 Answer: 5√2
  2. Simplify the following surd: √(75/3) A. 5 B. 4 C. 3 D. 6 Answer: 5
  3. What is the result of √20 - √5? A. 2√5 B. √15 C. 3√5 D. 4√5 Answer: 3√5
  4. If √a = 5 and √b = 7, what is the value of √(ab)? A. 40 B. 35 C. 42 D. 45 Answer: 35
  5. Simplify the surd: √(48/2) A. 4 B. 6 C. 8 D. 10 Answer: 4
  6. Perform the following operation on the surds: 3√27 - 2√12 A. 7√3 B. 6√3 C. 5√3 D. 4√3 Answer: 5√3
  7. If √p = 2 and √q = 3, what is the value of √(pq)? A. 6 B. 5 C. 7 D. 8 Answer: 6
  8. Simplify the surd: √(40/2) A. 4 B. 6 C. 8 D. 10 Answer: 4
  9. Evaluate the expression: √(121/2) + √(121/8) A. 11 B. 10 C. 12 D. 13 Answer: 12
  10. Find the value of √45 + √20. A. 7√5 B. 6√5 C. 5√5 D. 8√5 Answer: 7√5

Recommended Books

Past Questions

Wondering what past questions for this topic looks like? Here are a number of questions about Surds from previous years

Question 1 Report

The length of the line joining points (x,4) and (-x,3) is 7 units. Find the value of x.

Practice a number of Surds past questions