Surds and simple rational expressions
A clear starting point, worked steps and space to practise.
Start here
A rational expression is a fraction containing algebra. Cancel common factors, not separate terms.
A removed factor can hide an excluded value. Always keep the original domain restriction.
1 Work through these steps
- State values that make the original denominator zero; these are excluded.
- Factor numerator and denominator.
- Cancel complete common non-zero factors.
- Check the simplified expression keeps the original restrictions.
2 Follow a worked example
Simplify (x² − 9)/(x − 3).
- The original denominator excludes x = 3.
- Factor x² − 9 = (x − 3)(x + 3).
- Cancel the common factor: x + 3, with x ≠ 3.
Simplify √72.
- 72 = 36 × 2.
- √72 = √36 × √2.
- The result is 6√2.
More help with example 2
An exact surd keeps an irrational root in its exact form. Extract square factors to simplify square roots.
- Factor the number under the root.
- Find a perfect-square factor.
- Take that factor’s square root outside.
- Combine only like surds.
Strategy: Do not split a sum under a root: √(a + b) is not generally √a + √b.
3 Try it yourself
Show each step on paper or open Working out. Then check the answer.
-
Simplify (2x + 6)/2.
Check answer 1
Answer: x + 3
Factor 2(x + 3) in the numerator and divide the complete factor by 2.
-
Simplify √50 + √8.
Check answer 2
Answer: 7√2
√50 = 5√2 and √8 = 2√2; add the like surds.
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Check your understanding
Explain the worked example in your own words. Say why the method works, not just what the answer is.
Skills and explanation links
Key ideas taught here
- A rational expression is a fraction containing algebra. Cancel common factors, not separate terms.
- A removed factor can hide an excluded value. Always keep the original domain restriction.
Method and checking
Optional recap: related Maths skills
These earlier examples revisit background skills. Start with the topic example and practice above.
Useful vocabulary
- variable
- A letter or symbol representing a number.
- expression
- Numbers and symbols without an equals sign.
- equation
- A mathematical statement with an equals sign.
- solve
- Find the value that makes an equation true.
Rules for these related skills
Define what the letter stands for.
Do the same operation to both sides of an equation.
Put the answer back into the original expression or equation.
Evaluate 3n + 2 when n = 4.
- Substitute 4 for n.
- 3 x 4 + 2 = 12 + 2.
- Answer: 14.
Solve x + 7 = 19.
- Use the inverse operation.
- x = 19 - 7.
- x = 12.
Solve 3x - 5 = 16.
- Add 5 to both sides: 3x = 21.
- Divide by 3: x = 7.
Tickets cost $6 each plus a $4 booking fee. Write the cost for n tickets.
- n tickets cost 6n dollars.
- Add the booking fee.
- Cost = 6n + 4.
More visual models and everyday examples
Each everyday example has its own information and model. Use its stated values rather than copying values from another question.
Concept example. The story examples below have their own diagrams.
Keep both sides balanced at every step.
For parents and teachers
- Ask your child to explain the method aloud before writing the answer.
- Use a real-life context at home, such as shopping, cooking, sport scores, maps or timetables.
- Praise clear working and checking, not only speed.
- Ask your child to write the formula or rule first, then substitute values carefully.
Practise choosing a method from the information in the question. The examples here are Pi Leo Academy learning activities. Official NAPLAN testing takes place in Years 3, 5, 7 and 9; a topic appearing in these notes does not mean it is tested in NAPLAN.
These notes provide guided examples and practice; they do not certify completion of every curriculum requirement.