Surds (HSC SSCE Mathematics Advanced): Revision Notes

📚 Revision Notes
Revision NotesModel AnswersQuizzesFlashcards

Surds

Surds are a high-frequency exam topic in the SSCE Mathematics Advanced course. They often appear alongside algebraic fractions, so mastering this content will help you tackle a range of question types.

What is a surd?

A surd is a type of root (such as a square root) that cannot be expressed as a whole number or simple fraction. These are irrational numbers that remain in root form.

Examples of surds include:

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

These values cannot be simplified to give exact whole numbers or fractions, so they are left in their root form.

infoNote

Surds are different from perfect squares like 4\sqrt{4} or 9\sqrt{9}, which can be simplified to whole numbers (2 and 3 respectively). A surd remains in root form because it represents an irrational number.

Simplifying surds

When working with surds, you'll often need to simplify them to their most basic form. This involves extracting any perfect square factors from under the root sign.

Method for simplifying surds

To simplify a surd, follow these steps:

  1. Find the largest square factor of the number under the root
  2. Split the root into two separate roots
  3. Simplify by evaluating the perfect square
lightbulbExample

Worked Example: Simplifying 50\sqrt{50}

Let's simplify 50\sqrt{50} step by step.

Step 1: Find the square factor

50=25×250 = 25 \times 2

Step 2: Split the root

50=25×2\sqrt{50} = \sqrt{25} \times \sqrt{2}

Step 3: Simplify

=52= 5\sqrt{2}

Answer: 525\sqrt{2}

The key here is recognising that 2525 is a perfect square (52=255^2 = 25), so we can take it out from under the root as 55.

Multiplying surds

When multiplying surds together, you can combine them under a single root before simplifying.

Rule for multiplying surds

a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}

This means you multiply the numbers inside the roots together, then take the square root of the result.

lightbulbExample

Worked Example: Multiplying 3×12\sqrt{3} \times \sqrt{12}

Let's simplify 3×12\sqrt{3} \times \sqrt{12}.

Step 1: Multiply inside the root

=36= \sqrt{36}

Step 2: Simplify

=6= 6

Answer: 66

Since 3636 is a perfect square, the result is the whole number 66.

Adding and subtracting surds

When adding or subtracting surds, an important rule applies: you can only combine like surds.

What are like surds?

Like surds are surds that have the same number under the root sign. For example, 323\sqrt{2} and 525\sqrt{2} are like surds because they both contain 2\sqrt{2}.

infoNote

Think of this as being similar to combining like terms in algebra. Just as you can add 3x+5x=8x3x + 5x = 8x, you can add surds that have the same root.

lightbulbExample

Worked Example: Adding 32+523\sqrt{2} + 5\sqrt{2}

Let's simplify 32+523\sqrt{2} + 5\sqrt{2}.

Step 1: Combine like terms

=(3+5)2= (3 + 5)\sqrt{2}

Answer: 828\sqrt{2}

We add the coefficients (the numbers in front) while keeping the surd part the same.

Important note about unlike surds

chatImportant

Unlike surds cannot be simplified by addition or subtraction. For example:

2+3\sqrt{2} + \sqrt{3} cannot be simplified further

These surds have different numbers under the root, so they must remain separate.

Rationalising the denominator

Rationalising the denominator is the process of removing surds from the bottom of a fraction. In mathematics, we prefer not to leave surds in the denominator of a fraction.

Why rationalise?

Having a surd in the denominator can make calculations more difficult and answers less clear. Rationalising gives us a cleaner, more standard form that is easier to work with and is the expected format for final answers in exams.

Method for simple surds in the denominator

When you have a single surd in the denominator, multiply both the numerator and denominator by that same surd.

lightbulbExample

Worked Example: Rationalising 13\frac{1}{\sqrt{3}}

Let's rationalise 13\frac{1}{\sqrt{3}}.

Step 1: Multiply top and bottom by 3\sqrt{3}

=1×33×3= \frac{1 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}

=33= \frac{\sqrt{3}}{3}

Answer: 33\frac{\sqrt{3}}{3}

Note that 3×3=3\sqrt{3} \times \sqrt{3} = 3, which gives us a whole number in the denominator.

Method for binomial denominators

When the denominator contains a binomial expression (two terms) with a surd, we use the conjugate to rationalise. The conjugate is formed by changing the sign between the two terms.

For example, the conjugate of (2+3)(2 + \sqrt{3}) is (2−3)(2 - \sqrt{3}).

lightbulbExample

Worked Example: Rationalising 12+3\frac{1}{2 + \sqrt{3}}

Let's rationalise 12+3\frac{1}{2 + \sqrt{3}}.

Step 1: Multiply by the conjugate

The conjugate is (2−3)(2 - \sqrt{3}).

=1×(2−3)(2+3)(2−3)= \frac{1 \times (2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}

=2−3(2+3)(2−3)= \frac{2 - \sqrt{3}}{(2 + \sqrt{3})(2 - \sqrt{3})}

Step 2: Simplify the denominator using difference of squares

(2+3)(2−3)=4−3=1(2 + \sqrt{3})(2 - \sqrt{3}) = 4 - 3 = 1

Answer: 2−32 - \sqrt{3}

The denominator simplifies to 11, leaving us with just the numerator.

Common mistakes to avoid

chatImportant

Be aware of these frequent errors when working with surds:

  • Not simplifying fully: Leaving a surd like 50\sqrt{50} instead of simplifying it to 525\sqrt{2}
  • Adding unlike surds: Attempting to combine surds with different roots, such as 2+3\sqrt{2} + \sqrt{3}
  • Forgetting to rationalise: Leaving surds in the denominator of your final answer
  • Errors with conjugates: Making mistakes when multiplying binomials or using the difference of squares formula

Exam tips

infoNote

Here are some strategies to help you succeed with surds in exams:

  • Always look for square factors first when simplifying surds. Think about perfect squares: 4,9,16,25,36,49,64,1004, 9, 16, 25, 36, 49, 64, 100, etc.
  • Keep surds in their simplest form throughout your working and in your final answer.
  • For rationalising denominators:
    • If there's a single surd → multiply by the same surd
    • If there's a binomial → use the conjugate
  • Combine surds like algebra: Think of 32+52=823\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} as being similar to 3x+5x=8x3x + 5x = 8x
bookmarkSummary

Key Points to Remember:

  • Surds are irrational roots that cannot be expressed as whole numbers
  • Simplify surds by extracting perfect square factors
  • When multiplying surds, multiply the values inside the root: a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}
  • Only combine like surds (those with the same root)
  • Rationalise denominators by multiplying by the same surd (for simple cases) or the conjugate (for binomials)

Explore HSC SSCE Mathematics Advanced Model Answers by Topics

Algebraic Techniques

Solve Equations

Introduction to Functions and Relations

Linear Functions

Quadratic and Cubic Functions

Function Properties

Functions and Relations

Trigonometry

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

The Derivative

Exponential Functions

Logarithmic Functions

Transformations

Explore HSC SSCE Mathematics Advanced Quizzes by Topics

Algebraic Techniques

Solve Equations

Introduction to Functions and Relations

Linear Functions

Quadratic and Cubic Functions

Function Properties

Functions and Relations

Trigonometry

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

The Derivative

Exponential Functions

Logarithmic Functions

Transformations

Explore HSC SSCE Mathematics Advanced Flashcards by Topics

Algebraic Techniques

Solve Equations

Introduction to Functions and Relations

Linear Functions

Quadratic and Cubic Functions

Function Properties

Functions and Relations

Trigonometry

Trigonometric Identities and Equations

Probability

Data

Introduction to Rates of Change

The Derivative

Exponential Functions

Logarithmic Functions

Transformations

Join 100,000+ SSCE students studying Revision Notes with us.

Select your subjects, and get access to A+ resources today.