Surds
Working with irrational roots in exact form
๐ Introduction
When we take the square root of a number that's not a perfect square (like $\sqrt{2}$ or $\sqrt{3}$), we get an irrational number that can't be written as a simple fraction. These expressions are called surds, and they're essential in mathematics because they allow us to represent exact values rather than decimal approximations.
๐ฏ Learning Objectives
By the end of this lesson, you will be able to:
- Understand what surds are and why they're used
- Simplify surd expressions by taking out perfect square factors
- Add, subtract, multiply and divide surds
- Distinguish between rational and irrational numbers
๐ Key Concepts
What are Surds?
Surds are irrational numbers that are expressed using root symbols. They cannot be written as exact decimals or fractions because their decimal expansions go on forever without repeating.
๐ Definition
Surd: An irrational number that can be expressed as the root of a rational number. Common examples include $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$.
Not all roots are surds. For example, $\sqrt{4} = 2$ and $\sqrt{9} = 3$ are rational numbers, not surds, because they can be simplified to exact integers.
Rational vs. Irrational Numbers
๐ Definition
Rational Number: Any number that can be expressed as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
Irrational Number: Any real number that cannot be expressed as a fraction of integers.
Surds are a specific type of irrational number. Other famous irrational numbers include ฯ (pi) and e (Euler's number).
Simplifying Surds
A surd is in its simplest form when:
- The number under the root symbol has no perfect square factors (for square roots), or perfect cube factors (for cube roots), etc.
- There are no fractions under the root symbol
๐ก Pro Tip
To simplify a surd, look for the largest square number that divides into the number under the square root. For example, to simplify $\sqrt{12}$, notice that $12 = 4 \times 3$, and $4$ is a perfect square. So $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$.
Operations with Surds
Just like with other numbers, we can add, subtract, multiply, and divide surds.
๐ Rules for Surds
Addition/Subtraction: We can only add or subtract surds if they are "like surds" (have the same irrational part).
For example: $2\sqrt{5} + 3\sqrt{5} = 5\sqrt{5}$
Multiplication: $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$
Division: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
๐ Worked Examples
๐ Example 1: Simplifying Surds
Question: Simplify $\sqrt{20}$.
Solution:
Find the largest perfect square factor of 20.
The factors of 20 are: 1, 2, 4, 5, 10, 20
Of these, the largest perfect square is 4.
Express 20 as a product of this perfect square and another number.
Apply the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
๐ Example 2: Adding and Subtracting Surds
Question: Simplify $3\sqrt{12} - 2\sqrt{27}$.
Solution:
First, simplify each surd.
Now that both terms have the same surd part, we can perform the subtraction.
๐ช Practice Problems
๐ฏ Practice Questions
- Simplify $\sqrt{18}$
- Simplify $\sqrt{32} + \sqrt{50}$
- Simplify $\sqrt{8} \times \sqrt{6}$
- Simplify $\frac{4}{\sqrt{5}}$
๐ Show Solutions
1. $3\sqrt{2}$
2. $7\sqrt{2}$
3. $2\sqrt{12} = 4\sqrt{3}$
4. $\frac{4\sqrt{5}}{5}$
๐ Summary
๐ฏ Key Takeaways
- Surds are irrational numbers expressed as roots of rational numbers.
- Surds are used to give exact values rather than decimal approximations.
- To simplify a surd, find the largest perfect square factor and apply the rule $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$.
- You can only add or subtract surds if they have the same irrational part (like surds).
- When multiplying surds, use the rule $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$.
- When dividing surds, rationalize the denominator if necessary.