Understanding Surds

Definition: A surd is an irrational number that cannot be simplified to remove the square root (or cube root, etc.). Surds are left in root form because they represent exact values that can't be expressed as a simple fraction or decimal. For example, \\( \sqrt{2} \\) and \\( \sqrt{5} \\) are surds because their decimal expansions are non-terminating and non-repeating.

Why Surds Matter:

• Exactness: Surds provide exact values rather than approximations.
• Algebraic Manipulation: They are essential in solving equations and algebraic expressions where approximate values are not sufficient.

Basic Properties of Surds:

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

Worked Examples

Example 1: Simplifying Surds

Simplify \\( \sqrt{50} \\).

Solution: \\( \sqrt{50} = 5 \sqrt{2} \\).

Example 2: Multiplying Surds

Simplify \\( \sqrt{3} \times \sqrt{12} \\).

Solution: \\( \sqrt{3} \times \sqrt{12} = 6 \\).

Example 3: Rationalizing the Denominator

Rationalize the denominator of \\( \frac{5}{\sqrt{7}} \\).

Solution: \\( \frac{5}{\sqrt{7}} = \frac{5\sqrt{7}}{7} \\).

Example 4: Adding and Subtracting Surds

Simplify \\( 3\sqrt{5} + 2\sqrt{5} - \sqrt{5} \\).

Solution: \\( 4\sqrt{5} \\).

Example 5: Expanding Expressions Involving Surds

Expand \\( (2 + \sqrt{3})(3 - \sqrt{3}) \\).

Solution: \\( 3 + \sqrt{3} \\).

Practice Questions

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