Real numbers — surds and properties
A clear starting point, worked steps and space to practise.
Start here
An exact surd keeps an irrational root in its exact form. Extract square factors to simplify square roots.
Do not split a sum under a root: √(a + b) is not generally √a + √b.
1 Work through these steps
- Factor the number under the root.
- Find a perfect-square factor.
- Take that factor’s square root outside.
- Combine only like surds.
2 Follow a worked example
Simplify √72.
- 72 = 36 × 2.
- √72 = √36 × √2.
- The result is 6√2.
3 Try it yourself
Show each step on paper or open Working out. Then check the answer.
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Simplify √50 + √8.
Check answer 1
Answer: 7√2
√50 = 5√2 and √8 = 2√2; add the like surds.
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Working out
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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
- An exact surd keeps an irrational root in its exact form. Extract square factors to simplify square roots.
- Do not split a sum under a root: √(a + b) is not generally √a + √b.
Method and checking
Optional recap: related Maths skills
These earlier examples revisit background skills. Start with the topic example and practice above.
Useful vocabulary
- base
- The number being repeatedly multiplied.
- index
- The small raised number showing how many factors.
- power
- Another name for index notation.
- scientific notation
- A compact way to write very large or very small numbers.
Rules for these related skills
A positive whole-number index tells how many equal base factors are multiplied.
Only use index laws when their conditions are met.
For a non-zero base, any number to the power of 0 equals 1.
Write 2 x 2 x 2 x 2 using index notation.
- There are four factors of 2.
- Write 2 to the power of 4.
Simplify 3 squared x 3 cubed.
- Same base, so add indices.
- 2 + 3 = 5.
- Answer: 3 to the power of 5.
Write 4 500 000 in scientific notation.
- Move the decimal point after the first non-zero digit.
- 4.5 x 10 to the power of 6.
A bacteria count doubles each hour. If it starts at 100, write the amount after 3 hours.
- Doubling 3 times means multiply by 2 cubed.
- 100 x 2 cubed = 800.
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.
The index counts factors, not zeros.
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.