Surds, made simple
Learn how to simplify surds with the square-root bouncer analogy. Worked examples, exam-style questions, and a checklist to help you simplify and rationalise surds.
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Download the PDF (576 KB)Why does your calculator turn √12 into 2 √3, and why is that the same number?
Surds feel hard because they look messy. Think of the square-root bouncer who only lets square numbers out. This helps simplify roots into exact forms, not long decimals.
The picture to keep in your head: The square-root bouncer
Think of the square-root sign as a club with a bouncer. Only square numbers (4, 9, 16, 25...) are allowed out, and when one leaves it becomes its root: a 4 walks out as a 2. So √12 = √(4 × 3): the 4 leaves as 2, the 3 can't leave, so it's 2 √3. A surd is just a root that can't fully leave the club, so we keep it exact instead of writing a long decimal.
The method, every time
- Let the square numbers out first. Find the biggest square that divides the number under the root.
- Share the room with like surds. Only roots with the same number can be added or subtracted.
- Multiply roots together. When you multiply two roots, you can combine them under one square root.
- Clear the root from the bottom. Multiply top and bottom by the same root to make the denominator nice.
- Write the answer in its simplest form. Make sure no square numbers are left inside the root.
Words you'll see
- surd
- A root that can’t fully leave the club, so we keep it exact.
- simplify
- Make the surd look neater by letting square numbers out first.
- rationalise
- Remove the root from the bottom of a fraction by multiplying top and bottom.
Let the square numbers out
The square-root bouncer only lets square numbers out. So, split the number under the root into the biggest square number times what's left. Then the square number leaves as its root. This way, you simplify the surd properly.
Watch out
- Using a square number that isn't the biggest one, e.g. √72 = 2√18 and stopping. Always find the biggest square number first
- Writing √12 = 3√4. Check if the square number is actually a factor of the whole number
Roots can share a room
The square-root bouncer lets two roots share a room. When you multiply or divide roots, they merge into one. If the same root is multiplied, it becomes a square and walks out. Multiply the numbers outside and the roots inside. This is how you simplify expressions with surds.
Watch out
- Adding instead of multiplying the outside numbers. Multiply the numbers outside, not add them
- Forgetting to simplify the root at the end. Always check if the root can be simplified
Only like surds add up
The square-root bouncer only lets like surds out together. Think of 2√3 + 5√3 as 2 apples + 5 apples = 7 apples. But √2 + √3 is like apples and pears — they can’t be added. Always simplify surds first to see if they’re like terms.
Watch out
- Writing √2 + √3 = √5. You can’t add surds like numbers — they’re different types.
- Not simplifying each surd first. Always simplify surds before trying to add them.
Clear the root from the bottom
The square-root bouncer won't let a root stay at the bottom of a fraction. To tidy it up, you multiply top and bottom by the same root. That way, the root leaves the denominator and joins the numerator. This is called rationalising the denominator. You can only do this if the root is in the denominator.
Watch out
- Only multiplying the bottom. Multiply both top and bottom by the same root
- Not simplifying the final fraction. Divide numbers and simplify the result
23 exam-style surds questions
Show your working: most marks are for the method. Marks are a guide to how much working an exam would expect. Questions written by Bloom Education in the style of GCSE (AQA, Edexcel, OCR) and Cambridge IGCSE papers.
A · Simplify (Higher)
- 1Simplify √20.[2 marks]
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2√5
√20 = √(4 × 5). The 4 leaves as 2. So 2√5. - 2Simplify √45.[2 marks]
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3√5
√45 = √(9 × 5). The 9 leaves as 3. So 3√5. - 3Simplify √98.[2 marks]
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7√2
√98 = √(49 × 2). The 49 leaves as 7. So 7√2. - 4Simplify √300.[2 marks]
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10√3
√300 = √(100 × 3). The 100 leaves as 10. So 10√3. - 5Simplify √128.[2 marks]
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8√2
√128 = √(64 × 2). The 64 leaves as 8. So 8√2.
B · Multiply and divide (Higher)
- 6Simplify √5 × √5[2 marks]
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5
√5 × √5 = √(5 × 5) = √25 = 5 - 7Simplify 2√3 × 4√3[2 marks]
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24
2 × 4 = 8, √3 × √3 = √9 = 3, 8 × 3 = 24 - 8Simplify √6 × √10[2 marks]
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2√15
√(6 × 10) = √60 = √(4 × 15) = 2√15 - 9Simplify √48 ÷ √3[2 marks]
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4
√(48 ÷ 3) = √16 = 4 - 10Write √8 × √18 as an integer[2 marks]
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12
√(8 × 18) = √144 = 12
C · Add and subtract (Higher)
- 11Simplify 3√2 + 5√2[3 marks]
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8√2
Both terms have √2. Add the numbers in front: 3 + 5 = 8. So 8√2. - 12Simplify 4√3 + 2√12[3 marks]
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8√3
√12 = 2√3, so 2√12 = 4√3. Then 4√3 + 4√3 = 8√3. - 13Simplify 7√5 − 2√20[3 marks]
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3√5
√20 = 2√5, so 2√20 = 4√5. Then 7√5 - 4√5 = 3√5. - 14Simplify 2√18 − √8 + √32[3 marks]
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8√2
Simplify each root: √18 = 3√2, √8 = 2√2, √32 = 4√2. Now all terms have √2. Add and subtract: 2×3 = 6, 6 − 2 = 4, 4 + 4 = 8. So 8√2.
D · Expand brackets (Higher)
- 15Expand and simplify (√2 + 3)(√2 - 3)[3 marks]
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-7
Multiply using (a + b)(a - b) = a² - b². √2² - 3² = 2 - 9 = -7 - 16Expand and simplify (√5 + 2)2[3 marks]
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9 + 4√5
Use (a + b)2 = a² + 2ab + b². (√5)² + 2×√5×2 + 2² = 5 + 4√5 + 4 = 9 + 4√5 - 17Expand and simplify √3(2 + √3)[3 marks]
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3 + 2√3
Multiply each term: √3×2 + √3×√3 = 2√3 + 3 - 18Expand and simplify (√6 + √2)2[3 marks]
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8 + 4√3
(√6)² + 2√6√2 + (√2)² = 6 + 2√12 + 2 = 8 + 4√3, since √12 = 2√3.
E · Rationalise (Higher)
- 19Rationalise the denominator of 5/√5[3 marks]
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√5
Multiply numerator and denominator by √5. 5×√5 = 5√5. √5×√5 = 5. 5√5/5 = √5. - 20Rationalise the denominator of 12/√6[3 marks]
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2√6
Multiply numerator and denominator by √6. 12×√6 = 12√6. √6×√6 = 6. 12√6/6 = 2√6. - 21Rationalise the denominator of 1/(2√3)[3 marks]
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√3/6
Multiply numerator and denominator by √3. 1×√3 = √3. 2√3×√3 = 2×3 = 6. √3/6 = √3/6. - 22Rationalise the denominator of 4/(3 - √5)[3 marks]
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3 + √5
Multiply numerator and denominator by (3 + √5). (4×(3 + √5)) = 12 + 4√5. (3 - √5)(3 + √5) = 9 - 5 = 4. (12 + 4√5)/4 = 3 + √5. - 23Show that √75 = 5√3[3 marks]
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5√3
Split 75 into square and non-square parts. 75 = 25 × 3. √(25 × 3) = √25 × √3 = 5√3.
Surds exam checklist
- Biggest square number first. Always find the largest square that divides the number.
- Multiply roots together. Combine them under one root when multiplying.
- Add only like surds. Only roots with the same number can be added or subtracted.
- Multiply top and bottom when rationalising. Remove the root from the bottom of a fraction.
- Answer in its simplest form. Make sure no square numbers are inside the root.
- Check your work by expanding or substituting. Ensure your simplified form is the same as the original.
Got stuck on a question?
Find the part of this guide it belongs to and go through the worked example again, line by line. Then put your answer back into the question: if it doesn't work, look for a sign or a step you skipped.
All of this page as a 10-page guide, with every answer at the back. No sign-up.
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