Addition, Subtraction, Multiplication, and Rationalizing Surds

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1. Lesson Overview

Surds follow algebraic rules similar to variables like \(x\) and \(y\). You can add and subtract like surds, multiply and divide any surds, and rewrite fractions so that no square root remains in the denominator.

2. Definitions and Key Rules

A. Like Surds and Unlike Surds

Like Surds: Surds that have the exact same irrational root part once simplified to their simplest mixed form. (Examples: \(3\sqrt{2}\), \(5\sqrt{2}\), and \(-\sqrt{2}\))

Unlike Surds: Surds with different irrational root parts that cannot be combined by addition or subtraction. (Examples: \(4\sqrt{2}\) and \(4\sqrt{3}\))

B. Arithmetic Rules for Surds

  • Addition & Subtraction: \(x\sqrt{a} + y\sqrt{a} = (x + y)\sqrt{a}\)
  • Multiplication: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)  and  \(x\sqrt{a} \times y\sqrt{b} = xy\sqrt{ab}\)
  • Division: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
  • Squaring a Surd: \((\sqrt{a})^2 = \sqrt{a} \times \sqrt{a} = a\)

C. Rationalizing the Denominator

Definition: The process of converting a fraction with an irrational denominator (a surd) into an equivalent fraction with a rational denominator (a whole number).

Conjugate Surds: Two binomial surd expressions of the form \((a + \sqrt{b})\) and \((a - \sqrt{b})\) are conjugates of each other:

\[(a + \sqrt{b})(a - \sqrt{b}) = a^2 - (\sqrt{b})^2 = a^2 - b\]

3. Worked Examples (Step-by-Step)

Example 1: Addition and Subtraction of Surds

Simplify: \(\sqrt{50} + \sqrt{18} - \sqrt{8}\)

Solution:

Step 1: Simplify each entire surd into its simplest mixed surd form.
\[\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}\] \[\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}\] \[\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}\]
Step 2: Substitute the mixed surds back into the expression.
\[\sqrt{50} + \sqrt{18} - \sqrt{8} = 5\sqrt{2} + 3\sqrt{2} - 2\sqrt{2}\]
Step 3: Combine the coefficients of the like surd \(\sqrt{2}\).
\[= (5 + 3 - 2)\sqrt{2}\] \[= 6\sqrt{2}\]

Example 2: Multiplication and Expansion of Surds

Expand and simplify: \((3 + \sqrt{2})(4 - \sqrt{2})\)

Solution:

Step 1: Multiply each term in the first bracket by the second bracket.
\[(3 + \sqrt{2})(4 - \sqrt{2}) = 3(4 - \sqrt{2}) + \sqrt{2}(4 - \sqrt{2})\]
Step 2: Expand the brackets.
\[= (3 \times 4) - (3 \times \sqrt{2}) + (\sqrt{2} \times 4) - (\sqrt{2} \times \sqrt{2})\] \[= 12 - 3\sqrt{2} + 4\sqrt{2} - 2\]
Step 3: Group the rational numbers together and the like surds together.
\[= (12 - 2) + (-3\sqrt{2} + 4\sqrt{2})\] \[= 10 + \sqrt{2}\]

Example 3: Rationalizing a Monomial Denominator

Express \(\frac{10}{\sqrt{5}}\) with a rational denominator.

Solution:

Step 1: Multiply both the numerator and the denominator by \(\sqrt{5}\).
\[\frac{10}{\sqrt{5}} = \frac{10}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}}\]
Step 2: Simplify the numerator and denominator separately.
\[= \frac{10\sqrt{5}}{(\sqrt{5})^2}\] \[= \frac{10\sqrt{5}}{5}\]
Step 3: Divide \(10\) by \(5\) to obtain the final answer.
\[= 2\sqrt{5}\]

Example 4: Rationalizing a Binomial Denominator (O/L Exam Standard)

Express \(\frac{6}{\sqrt{7} - 2}\) with a rational denominator.

Solution:

Step 1: Identify the conjugate of the denominator \((\sqrt{7} - 2)\), which is \((\sqrt{7} + 2)\).
Step 2: Multiply both the numerator and denominator by \((\sqrt{7} + 2)\).
\[\frac{6}{\sqrt{7} - 2} = \frac{6}{\sqrt{7} - 2} \times \frac{\sqrt{7} + 2}{\sqrt{7} + 2}\]
Step 3: Apply the identity \((a - b)(a + b) = a^2 - b^2\) to the denominator.
\[= \frac{6(\sqrt{7} + 2)}{(\sqrt{7})^2 - 2^2}\] \[= \frac{6(\sqrt{7} + 2)}{7 - 4}\]
Step 4: Simplify the denominator and cancel common factors.
\[= \frac{6(\sqrt{7} + 2)}{3}\] \[= 2(\sqrt{7} + 2)\] \[= 2\sqrt{7} + 4\]

4. Practice Exercises for Students

  1. Simplify: \(3\sqrt{12} + 2\sqrt{27} - \sqrt{75}\)
  2. Expand and simplify: \((\sqrt{5} + 2)(\sqrt{5} - 2)\)
  3. Rationalize the denominator of each fraction:
    1. \(\frac{12}{\sqrt{3}}\)
    2. \(\frac{8}{3 - \sqrt{5}}\)
Show Answers

1. \(7\sqrt{3}\)   (\(6\sqrt{3} + 6\sqrt{3} - 5\sqrt{3}\))

2. \(1\)   (\(5 - 4\))

3. (a) \(4\sqrt{3}\)    (b) \(2(3 + \sqrt{5}) = 6 + 2\sqrt{5}\)

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