Simplifying Expressions Containing Surds

Fundamental Rules of Surds

• Multiplication Rule for Surds:

  • When multiplying two surds, multiply the numbers inside the root signs (radicands) together:     $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

  • Example:     $\sqrt{5} \times \sqrt{6} = \sqrt{5 \times 6} = \sqrt{30}$

• Division Rule for Surds:

  • When dividing two surds, divide the numbers inside the root signs. This can be expressed in division notation or as a fraction:     $\sqrt{a} \div \sqrt{b} = \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

  • Example:     $\sqrt{20} \div \sqrt{10} = \frac{\sqrt{20}}{\sqrt{10}} = \sqrt{\frac{20}{10}} = \sqrt{2}$

• Restrictions on Addition and Subtraction with Different Radicands:

  • Surds with different numbers inside the root signs cannot be combined via addition or subtraction:     $\sqrt{13} + \sqrt{6} \neq \sqrt{19}$     $\sqrt{13} - \sqrt{6} \neq \sqrt{7}$

• Addition and Subtraction Rules for Surds with Matching Radicands:

  • When surds share the exact same number inside the radical, combine them by adding or subtracting their coefficients (the numbers in front of the surd):

  • Addition Example:       $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$

  • Subtraction Example:       $6\sqrt{7} - 2\sqrt{7} = 4\sqrt{7}$

• Self-Multiplication Rule:

  • Multiplying a surd by itself removes the radical sign, leaving only the radicand:     $\sqrt{a} \times \sqrt{a} = a$

  • Conceptual Explanation:     $\sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7$

Standard Exam Notation and Simplification Strategy

• Standard Answer Format:

  • Exam problems frequently require writing a simplified surd expression in the form:     $a + b\sqrt{5}$

  • The variables $a$ and $b$ represent integers (positive or negative whole numbers).

  • Valid example expressions in this format include:     $4 + 3\sqrt{5}$     $7 - 2\sqrt{5}$

• Core Simplification Method:

  • Rewrite the surd as a product of its factor pairs.

  • Select the factor pair that contains the largest square number.

  • Take the square root of the square factor and multiply it by the remaining surd factor.

Worked Example 1: Simplifying Complex Expressions

• Problem Statement:

  • Simplify the expression:     $\sqrt{125} - 2\sqrt{45} + (\sqrt{5} + 2)^2$

  • Write the final answer in the form $a + b\sqrt{5}$, where $a$ and $b$ are integers.

• Step 1: Simplifying the First Term ($\sqrt{125}$):

  • Identify factor pairs of $125$ with the largest square number: $25$ and $5$.

  • Rewrite as:     $\sqrt{125} = \sqrt{25} \times \sqrt{5}$

  • Evaluate the square root of $25$ ($\sqrt{25} = 5$):     $5 \times \sqrt{5} = 5\sqrt{5}$

• Step 2: Simplifying the Second Term ($-2\sqrt{45}$):

  • Express $\sqrt{45}$ using its factor pair containing the largest square number ($9$ and $5$):     $\sqrt{45} = \sqrt{9} \times \sqrt{5}$

  • Evaluate the square root of $9$ ($\sqrt{9} = 3$):     $\sqrt{45} = 3\sqrt{5}$

  • Include the initial coefficient of $2$:     $2 \times 3 \times \sqrt{5} = 6\sqrt{5}$

  • Include the subtraction operator from the original expression:     $-6\sqrt{5}$

• Step 3: Simplifying the Third Term ($(\sqrt{5} + 2)^2$):

  • A squared binomial indicates multiplying the entire bracketed term by itself:     $(\sqrt{5} + 2)(\sqrt{5} + 2)$

  • Expand the brackets by multiplying each term in the first bracket by each term in the second bracket:

  • Multiply first terms:       $\sqrt{5} \times \sqrt{5} = 5$

  • Multiply outer terms:       $\sqrt{5} \times 2 = 2\sqrt{5}$

  • Multiply inner terms:       $2 \times \sqrt{5} = 2\sqrt{5}$

  • Multiply last terms:       $2 \times 2 = 4$

  • Combine the numeric values and like surd terms:     $5 + 4 = 9$     $2\sqrt{5} + 2\sqrt{5} = 4\sqrt{5}$

  • Simplified form of the third term:     $9 + 4\sqrt{5}$

• Step 4: Combining All Simplified Terms:

  • Sum the simplified results from all three terms:     $5\sqrt{5} - 6\sqrt{5} + 9 + 4\sqrt{5}$

  • Group and combine like surd terms:     $(5 - 6 + 4)\sqrt{5} + 9 = 3\sqrt{5} + 9$

• Step 5: Formatting for Required Exam Form:

  • Rearrange $3\sqrt{5} + 9$ to fit the target format $a + b\sqrt{5}$:     $9 + 3\sqrt{5}$

  • Corresponding integer values are $a = 9$ and $b = 3$.

Worked Example 2: Alternative Simplification Problem

• Problem Statement:

  • Simplify the expression:     $\sqrt{48} + 2\sqrt{75} + (\sqrt{3})^2$

  • Express the final answer in the form $a + b\sqrt{3}$.

• Step 1: Simplify $\sqrt{48}$:

  • Identify factor pair with the largest square number ($16$ and $3$):     $\sqrt{48} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3}$

• Step 2: Simplify $2\sqrt{75}$:

  • Identify factor pair for $75$ with the largest square number ($25$ and $3$):     $\sqrt{75} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$

  • Multiply by coefficient $2$:     $2 \times 5\sqrt{3} = 10\sqrt{3}$

• Step 3: Simplify $(\sqrt{3})^2$:

  • Apply self-multiplication rule:     $(\sqrt{3})^2 = 3$

• Step 4: Sum and Format Solution:

  • Combine all simplified terms:     $4\sqrt{3} + 10\sqrt{3} + 3 = 14\sqrt{3} + 3$

  • Rearrange to match target form $a + b\sqrt{3}$:     $3 + 14\sqrt{3}$

  • Corresponding integer values are $a = 3$ and $b = 14$.