Surds

Surds Lesson

Lesson Objectives

By the end of this lesson, students should be able to:

Define surds and distinguish them from rational and irrational numbers.
Simplify simple and compound surds.
Add, subtract, multiply, and divide surds.
Rationalize the denominator involving surds.

Introduction

A surd is an irrational number that cannot be simplified to remove the root symbol entirely. For example, \(\sqrt{2}\) is a surd, but \(\sqrt{4} = 2\) is not.

Core Lesson Content

Key laws of surds:

\(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
\(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
\((\sqrt{a})^2 = a\)

Simplification of Surds

Example 1: Simplify \(\sqrt{72}\).
\(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)

Answer: \(6\sqrt{2}\)

Example 2: Simplify \(2\sqrt{3} + 5\sqrt{3}\).
\((2 + 5)\sqrt{3} = 7\sqrt{3}\)

Answer: \(7\sqrt{3}\)

Multiplication and Division

Example 3: Multiply \(\sqrt{2} \times \sqrt{18}\).
\(= \sqrt{36} = 6\)

Answer: 6

Example 4: Divide \(\frac{\sqrt{27}}{\sqrt{3}}\).
\(= \sqrt{9} = 3\)

Answer: 3

Rationalization

Example 5: Rationalize \(\frac{5}{\sqrt{2}}\).
\(= \frac{5\sqrt{2}}{2}\)

Answer: \(\frac{5\sqrt{2}}{2}\)

Example 6: Rationalize \(\frac{3}{1 - \sqrt{2}}\).
Multiply by conjugate: \(\frac{3(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})} = \frac{3(1 + \sqrt{2})}{-1} = -3 - 3\sqrt{2}\)

Answer: \(-3 - 3\sqrt{2}\)

Compound Surds

Example 7: Simplify \(\sqrt{12} + \sqrt{27}\).
\(= 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}\)

Answer: \(5\sqrt{3}\)

Example 8: Simplify \(\sqrt{20} - \sqrt{5}\).
\(= 2\sqrt{5} - \sqrt{5} = \sqrt{5}\)

Answer: \(\sqrt{5}\)

Using Surds in Algebra

Example 9: If \(a = \sqrt{2}, b = \sqrt{8}\), find \(a + b\).
\(b = \sqrt{8} = 2\sqrt{2}\), so \(a + b = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}\)

Answer: \(3\sqrt{2}\)

Example 10: Expand \((\sqrt{2} + 1)^2\).
\(= (\sqrt{2})^2 + 2 \cdot \sqrt{2} \cdot 1 + 1^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2}\)

Answer: \(3 + 2\sqrt{2}\)

Exercises

Simplify \(\sqrt{72}\)
Simplify \(2\sqrt{3} + 5\sqrt{3}\)
Multiply \(\sqrt{2} \times \sqrt{18}\)
Rationalize \(\frac{5}{\sqrt{2}}\)
Evaluate \(\sqrt{12} + \sqrt{27}\)
[WAEC] Simplify \(\sqrt{75} - \sqrt{12}\) (Past Question)
Simplify \(\frac{\sqrt{3}}{\sqrt{5}}\)
[NECO] Rationalize and simplify \(\frac{2}{1 + \sqrt{3}}\) (Past Question)
[JAMB] If \(a = \sqrt{2}, b = \sqrt{8}\), simplify \(a + b\) (Past Question)
Expand \((\sqrt{2} + 1)^2\)

Conclusion / Recap

Surds are roots that cannot be expressed as exact rational numbers. In this lesson, we have explored simplification, rationalization, and operations on surds. Mastery of surds is crucial for progressing to logarithms and advanced algebra.