Dealing With Surds Revision Notes for Leaving Cert Mathematics

Dealing With Surds

What are rational and irrational numbers?

Understanding the difference between rational and irrational numbers is essential before working with surds.

Rational numbers are any numbers that can be written as a fraction in the form $\frac{a}{b}$ , where both $a$ and $b$ are integers and $b \neq 0$ . These numbers have decimal representations that either terminate or repeat in a pattern.

Examples of rational numbers include:

$3 = \frac{3}{1}$
$\frac{2}{3}$
$0.45 = \frac{45}{100}$
$-\frac{3}{4}$
$1\frac{3}{8} = \frac{11}{8}$

Irrational numbers are numbers that cannot be written as exact fractions. When you use a calculator to find $\sqrt{2}$ , you get 1.41421362... This decimal never ends and never repeats in a pattern. Such numbers are called irrational numbers.

Understanding surds

Surds are a special type of irrational number. They are square root expressions where the number under the square root sign does not have an exact whole number value.

infoNote

Examples of surds include $\sqrt{3}$ , $\sqrt{5}$ , $\sqrt{15}$ , and so on. These cannot be simplified to give exact decimal values.

Simplifying surds to their simplest form

The key to working with surds is being able to simplify them using these fundamental properties:

chatImportant

Essential Surd Properties:

$\sqrt{ab} = \sqrt{a} \times \sqrt{b}$

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

These properties are the foundation for all surd simplification.

Let's see how this works in practice. Consider $\sqrt{100} = 10$ . We can also write this as: $\sqrt{100} = \sqrt{25 \times 4} = \sqrt{25} \times \sqrt{4} = 5 \times 2 = 10$

We can use this property to simplify surds where possible:

$\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
$\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}$
$\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$

The expression $2\sqrt{2}$ is said to be the simplest form of $\sqrt{8}$ because we have taken out the largest possible perfect square factor.

Adding and subtracting surds

Surds can only be added or subtracted when they have the same irrational parts (like terms). If the irrational parts are not the same, we must first reduce each surd to its simplest form where possible.

lightbulbExample

Worked Example 1: Simplify $\sqrt{5} + \sqrt{45} - \sqrt{20}$

First, we express each surd in its simplest form:

$\sqrt{5}$ is already in simplest form
$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$
$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

Now we can combine like terms: $\sqrt{5} + \sqrt{45} - \sqrt{20} = \sqrt{5} + 3\sqrt{5} - 2\sqrt{5} = (1 + 3 - 2)\sqrt{5} = 2\sqrt{5}$

Multiplying surds

When multiplying surds, we multiply the rational factors (whole numbers) separately from the irrational factors (the square roots).

Examples:

$\sqrt{6} \times \sqrt{2} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$2\sqrt{3} \times 3\sqrt{5} = (2 \times 3) \times (\sqrt{3} \times \sqrt{5}) = 6\sqrt{15}$
$\sqrt{32} \times \sqrt{48} = \sqrt{16 \times 2} \times \sqrt{16 \times 3} = 4\sqrt{2} \times 4\sqrt{3} = 16\sqrt{6}$

infoNote

Remember the useful fact: $\sqrt{6} \times \sqrt{6} = 6$

lightbulbExample

Worked Example 2: Simplify $(2\sqrt{5} - 3)(2\sqrt{5} + 3)$

Using the difference of squares pattern $(a-b)(a+b) = a^2 - b^2$ : $(2\sqrt{5} - 3)(2\sqrt{5} + 3) = (2\sqrt{5})^2 - 3^2 = 4(5) - 9 = 20 - 9 = 11$

Solving equations involving surds

When solving equations containing square roots, it's crucial to follow a systematic approach to avoid errors.

chatImportant

Essential Steps for Solving Surd Equations:

Isolate the square root on one side of the equation
Square both sides to eliminate the square root symbol
Solve the resulting equation
Check your solutions by substituting back into the original equation

The checking step is crucial because squaring both sides can sometimes introduce false solutions.

lightbulbExample

Worked Example 3: Solve $2 + \sqrt{4x - 3} = x$

Step 1: Isolate the square root term $2 + \sqrt{4x - 3} = x$ $\sqrt{4x - 3} = x - 2$

Step 2: Square both sides $4x - 3 = (x - 2)^2$ $4x - 3 = x^2 - 4x + 4$

Step 3: Rearrange and solve $0 = x^2 - 4x + 4 - 4x + 3$ $0 = x^2 - 8x + 7$ $0 = (x - 7)(x - 1)$

So $x = 7$ or $x = 1$

Step 4: Check solutions in the original equation

When $x = 7$ : $2 + \sqrt{25} = 2 + 5 = 7$ ✓ Correct
When $x = 1$ : $2 + \sqrt{1} = 2 + 1 = 3 \neq 1$ ✗ Incorrect

Therefore, the correct solution is $x = 7$ .

chatImportant

Critical Note: The square root of a number is taken as the positive value only. So $\sqrt{25} = 5$ (not ±5). This is why checking solutions is essential when solving surd equations.

bookmarkSummary

Key Points to Remember:

Rational numbers can be written as fractions; irrational numbers cannot be written as exact fractions
Surds are square roots of numbers that don't have exact whole number values
Use $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ to simplify surds to their simplest form
Like surds (same square root part) can be added or subtracted
When solving surd equations, always isolate the square root, square both sides, then check your answers

Dealing With Surds (Leaving Cert Mathematics): Revision Notes