Adding and Subtracting Surds

Surds are integral for achieving exact values in calculations, which are crucial in mathematical proofs and operations. This revision note delves into surds, emphasising their importance, strategies for simplification, and operations involving addition and subtraction.

Definition of Surds

Surds: are irrational numbers expressible as the square root of a non-perfect square. They yield exact values in calculations.

Importance of Exactness

• Mathematical Precision: Surds provide exact values, essential in proofs and algebraic operations where approximations might lead to inaccuracies.

Geometric Illustration

Geometry often involves surds. For example, the diagonal of a square with side 1 measures $\sqrt{2}$.

In right triangles, the Pythagorean theorem frequently highlights surds by calculating the hypotenuse as a surd.

Common Surds

• Examples:
  • $\sqrt{2}$
  • $\sqrt{3}$
  • $\sqrt{5}$

These commonly arise in mathematical problems, such as solving $x^2 = 2$, resulting in $x = \pm \sqrt{2}$.

Introduction to Index Laws

Index Laws: Rules that handle expressions with powers, crucial for simplifying surds.

• Importance: They simplify challenging surd expressions, facilitating easier calculations.

Key Index Laws

• Product of Powers: $a^m \times a^n = a^{m+n}$
  • Example: $2^2 \times 2^3 = 2^5$
• Quotient of Powers: $\frac{a^m}{a^n} = a^{m-n}$
  • Example: $\frac{3^5}{3^2} = 3^3$
• Power of a Product: $(ab)^m = a^m \times b^m$
  • Example: $(3 \times 4)^2 = 3^2 \times 4^2$
• Power of a Quotient: $\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}$
  • Example: $\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3}$

Application of Index Laws with Surds

• Example: Simplify $\sqrt{a} \times \sqrt{a} = a$.
• Verify: Ensure terms under the square root are like before simplifying.

Simplification of Surds

Simplifying surds clarifies expressions and is essential for operations.

Prime Factorisation Method

• Definition: Prime factorisation breaks down a number into its prime components, aiding in identifying square factors in surds.
• Formula Application: $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ to simplify surds.

Worked Examples

• Example 1: Simplify $\sqrt{50}$

  • Factorise: $50 = 25 \times 2$
  • Simplify: $\sqrt{50} \rightarrow \sqrt{25 \times 2} \rightarrow 5\sqrt{2}$

• Example 2: Simplify $\sqrt{72}$

  • Factorise: $72 = 36 \times 2$
  • Simplify: $\sqrt{72} \rightarrow \sqrt{36 \times 2} \rightarrow 6\sqrt{2}$
  infoNote

  Tip: Always verify breakdowns thoroughly for complete simplification.

Visualisation and Tools

Factor Tree Diagrams

• Illustrate the breakdown of numbers visually using factor trees to simplify surds effectively.

Adding and Subtracting Like Surds

Like Surds: Surds sharing the same radicand that can be directly added or subtracted.

• Examples
  • $\sqrt{3}$
  • $2\sqrt{3}$
  • $5\sqrt{3}$

Procedure for Operations

• Identify like surds by comparing radicands.

• Add or subtract the coefficients, keeping the radicand constant.

• Example: $3\sqrt{5} + 2\sqrt{5} = (3+2)\sqrt{5} = 5\sqrt{5}$.

Practice Exercises

• Calculate: $4\sqrt{3} - \sqrt{3} = 3\sqrt{3}$
• Identify like surds from: $2\sqrt{2}, 5\sqrt{7}, \sqrt{2}, 4\sqrt{7}$
  • Solution: $2\sqrt{2}$ and $\sqrt{2}$ are like surds; $5\sqrt{7}$ and $4\sqrt{7}$ are like surds.

Simplification of Unlike Surds

Introduction to Unlike Surds

Simplification Process

• Purpose: Determine if unlike surds can become like surds for combination.
• Methods:
  • Prime factorisation: Decompose each radicand.
  • Perfect square factors: Simplify using perfect squares.

Example

• Simplify $\sqrt{18}$ and $\sqrt{8}$.
  • Prime Factorise: $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$ and $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$.
  • Combine: Result: $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$

Common Errors

• Frequent Mistakes: Not fully simplifying, adding unlike surds.

Practice Exercises

• Simplify and Combine: $\sqrt{12} + \sqrt{27}$
  • Solution: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$ and $\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$
  • Therefore, $\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$
• Simplify Difference: $2\sqrt{50} - \sqrt{18}$
  • Solution: $2\sqrt{50} = 2\sqrt{25 \times 2} = 2 \times 5\sqrt{2} = 10\sqrt{2}$ and $\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$
  • Therefore, $2\sqrt{50} - \sqrt{18} = 10\sqrt{2} - 3\sqrt{2} = 7\sqrt{2}$
• Combinability Test: $\sqrt{5} + \sqrt{20}$
  • Solution: $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$
  • Therefore, $\sqrt{5} + \sqrt{20} = \sqrt{5} + 2\sqrt{5} = 3\sqrt{5}$

Rationalising the Denominator

Rationalising the Denominator: Eliminate surds to simplify expressions, facilitating arithmetic and other operations.

• Objective: Simplify expressions for easier calculations.

Step-by-Step Process

• Identify the Surd in the denominator.
• Conjugates and Multiplication:
  • For single surds: Multiply by the surd itself.
  • For multi-term denominators: Use conjugates.

Example 1: Simplify $\frac{1}{\sqrt{2}}$.

• Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ to obtain $\frac{\sqrt{2}}{2}$.

Example 2: Simplify $\frac{3}{2+\sqrt{3}}$.

• Multiply by $\frac{2-\sqrt{3}}{2-\sqrt{3}}$ to simplify.

Practice Problems

• Rationalise $\frac{5}{\sqrt{7}}$.
  • Solution: $\frac{5}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{5\sqrt{7}}{7}$
• Simplify $\frac{4}{1+\sqrt{5}}$.
  • Solution: $\frac{4}{1+\sqrt{5}} \times \frac{1-\sqrt{5}}{1-\sqrt{5}} = \frac{4(1-\sqrt{5})}{(1+\sqrt{5})(1-\sqrt{5})} = \frac{4(1-\sqrt{5})}{1-5} = \frac{4(1-\sqrt{5})}{-4} = -1+\sqrt{5}$

Definition and Importance of Mixed Expressions

Mixed Expressions: Algebraic expressions composed of both rational numbers and surds.

• Significance: Distinguish components for precise simplification.
• Practical Implications: Accurate solutions.

Detailed Strategies for Simplification

Identify and Isolate

• Step 1: Separate rational numbers and surd components.
• Step 2: Simplify individually.

Example

• Expression: $3\sqrt{2} + \sqrt{18}$.
• Simplification: Convert to like surds: $3\sqrt{2} + 3\sqrt{2} = 6\sqrt{2}$.

Dealing with Unlike Surds

• Necessity: Simplify differing surds.

Practice Problem-Solving Exercises

• Exercise 1: Simplify: $2 + \sqrt{18} + \sqrt{50}$.
  • Simplify each surd: $\sqrt{18} = 3\sqrt{2}$ and $\sqrt{50} = 5\sqrt{2}$
  • Combine: Result: $2 + 3\sqrt{2} + 5\sqrt{2} = 2 + 8\sqrt{2}$.

Mastering these techniques enables students to approach advanced problems with confidence.