Surds

Introduction

In mathematics, surds are a type of irrational number expressed as $\sqrt{n}$, where $n$ is a non-square number. Surds are common in various mathematical calculations, including algebra, trigonometry, and geometry. Understanding how to manipulate surds is crucial for solving complex mathematical problems.

Simplifying Surds

Definition:

A surd is a square root that cannot be simplified to give a whole number. For example, $\sqrt{2}$ and $\sqrt{5}$ are surds.

Example:

Simplify the surd $\sqrt{18}$.

Solution: $$ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} $$

Adding and Subtracting Surds

Definition:

When adding or subtracting surds, ensure that the surds have the same root and then combine like terms.

Example:

Simplify $\sqrt{12} + \sqrt{27}$.

Solution: $$ \sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3} $$

Multiplying Surds

Definition:

To multiply surds, multiply the numbers outside the square root and then multiply the numbers inside the square root.

Example:

Multiply $\sqrt{5}$ by $\sqrt{8}$.

Solution: $$ \sqrt{5} \times \sqrt{8} = \sqrt{5 \times 8} = \sqrt{40} = 2\sqrt{10} $$

Dividing Surds

Definition:

To divide surds, divide the numbers outside the square roots and then divide the numbers inside the square roots.

Example:

Divide $\sqrt{15}$ by $\sqrt{3}$.

Solution: $$ \frac{\sqrt{15}}{\sqrt{3}} = \sqrt{\frac{15}{3}} = \sqrt{5} $$

Rationalizing Surds

Definition:

Rationalizing the denominator of a fraction means getting rid of any square roots from the denominator.

Example:

Rationalize the denominator of $\frac{1}{\sqrt{7}}$.

Solution: $$ \frac{1}{\sqrt{7}} = \frac{1 \times \sqrt{7}}{\sqrt{7} \times \sqrt{7}} = \frac{\sqrt{7}}{7} $$

Common Mistakes

• Forgetting to simplify surds fully.
• Adding or subtracting surds with different roots.
• Incorrectly multiplying or dividing surds.

Key Points

• Surds are irrational numbers expressed as square roots.
• When adding or subtracting surds, ensure they have the same root.
• To multiply surds, multiply the numbers outside and inside the square roots separately.
• To divide surds, divide the numbers outside and inside the square roots separately.
• Rationalizing surds involves removing square roots from the denominator.

Practice Questions

1. Simplify $\sqrt{27}$.
2. Evaluate $\sqrt{72} - \sqrt{18}$.
3. Multiply $\sqrt{10}$ by $\sqrt{20}$.
4. Divide $\sqrt{24}$ by $\sqrt{6}$.
5. Rationalize the denominator of $\frac{1}{\sqrt{3}}$.

Worked Answers

1. Solution: $$ \sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3} $$

2. Solution: $$ \sqrt{72} - \sqrt{18} = 6\sqrt{3} - 3\sqrt{2} $$

3. Solution: $$ \sqrt{10} \times \sqrt{20} = \sqrt{10 \times 20} = \sqrt{200} = 10\sqrt{2} $$

4. Solution: $$ \frac{\sqrt{24}}{\sqrt{6}} = \sqrt{4} = 2 $$

5. Solution: $$ \frac{1}{\sqrt{3}} = \frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{\sqrt{3}}{3} $$