Surds
Surds: simplifying, adding and expanding
In this note
- 1.18Write a surd in its simplest form, e.g. \(\sqrt{300}=10\sqrt3\).
- 1.18Add, subtract, multiply and divide surds.
- 1.18Expand brackets with surds into the form \(a+b\sqrt c\).
Before you start: square numbers and expanding brackets. Next: rationalising the denominator.
Formula bank
What a surd is
A surd is a square root that does not come out as a whole number or fraction: \(\sqrt2\), \(\sqrt{12}\). Writing \(\sqrt2\) rather than 1.414… keeps the value exact. Surds behave like letters in algebra: like surds add, and \(\sqrt a\times\sqrt a=a\).
Simplifying a surd
General methodHow to simplify any surd
- Find the LARGEST square factor of the number under the root (4, 9, 16, 25, 36, 49, 64, 81, 100, …).
- Split: \(\sqrt{300}=\sqrt{100\times3}=\sqrt{100}\times\sqrt3\).
- Root the square: \(10\sqrt3\).
- Check nothing under the root still has a square factor.
Example Simplify \(\sqrt{72}\).
Largest square factor of 72 is 36.
- \(\sqrt{36\times2}=\sqrt{36}\times\sqrt2\)
Answer \(6\sqrt2\)
Try it yourself
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Answer \(3\sqrt5\)
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Answer \(7\sqrt2\)
Hint
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Answer \(4\sqrt5\)
Adding and subtracting
General methodHow to add or subtract surds
- Simplify EVERY surd first.
- Like surds (same number under the root) add like like terms: \(5\sqrt5-2\sqrt5=3\sqrt5\).
- Unlike surds (\(\sqrt2+\sqrt3\)) stay as they are.
Example Simplify \(\sqrt{125}-\sqrt{20}\).
Simplify both; they become multiples of √5.
- \(\sqrt{125}=5\sqrt5\), \(\sqrt{20}=2\sqrt5\)
- \(5\sqrt5-2\sqrt5\)
Answer \(3\sqrt5\)
Try it yourself
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Answer \(5\sqrt3\)
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Answer \(\sqrt7\)
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Answer \(6\sqrt2\)
Multiplying and expanding
General methodHow to multiply surds and expand brackets
- Multiply numbers with numbers and roots with roots: \(2\sqrt5\times3\sqrt{10}=6\sqrt{50}\), then simplify.
- Brackets: multiply every term by every term, using \(\sqrt a\times\sqrt a=a\).
- Collect whole numbers and surd terms: answer in the form \(a+b\sqrt c\).
Example Expand and simplify \((2\sqrt5-1)(3-\sqrt5)\).
Four products, then collect.
- \(6\sqrt5-2\times5-3+\sqrt5\)
- \(=6\sqrt5-10-3+\sqrt5\)
Answer \(-13+7\sqrt5\)
Try it yourself
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Answer \(30\sqrt2\)
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Answer 13
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Answer \(11+4\sqrt7\)
Ways they test it
Type 1 Simplify a sum or difference
"Simplify \(\sqrt{175}-\sqrt{28}\)." (2 marks)
Spot it two surds with + or −.
- Simplify each (M1).
- Collect like surds (A1).
For example \(\sqrt{175}-\sqrt{28}\)
- \(5\sqrt7-2\sqrt7\)
Answer \(3\sqrt7\)
Type 2 Expand into a + b√c
"Expand \((3+\sqrt2)(1+4\sqrt2)\). Give your answer in the form \(a+b\sqrt2\)."
Spot it brackets and "in the form".
- Four products.
- Collect.
For example \((3+\sqrt2)(1+4\sqrt2)\)
- \(3+12\sqrt2+\sqrt2+8\)
Answer \(11+13\sqrt2\)
Type 3 Find k
"\(\sqrt{40}\times\sqrt8=k\sqrt5\). Find k."
Spot it "where k is an integer".
- Multiply under one root.
- Simplify to the given surd.
For example \(\sqrt{320}\)
- \(\sqrt{64\times5}\)
Answer \(k=8\)
Mistakes to avoid
- ✗ \(\sqrt{125}-\sqrt{20}=\sqrt{105}\)✓ \(3\sqrt5\)There is no rule for the root of a difference.
- ✗ \(\sqrt{72}=2\sqrt{18}\) (final)✓ \(6\sqrt2\)Use the largest square factor.
- ✗ \((\sqrt7+2)^{2}=11\)✓ \(11+4\sqrt7\)Don't forget the middle term.
- ✗ \(\sqrt2\times\sqrt2=\sqrt4\) left✓ 2Finish it.
Summary
- Simplify: take out the largest square factor.
- \(\sqrt{ab}=\sqrt a\sqrt b\); \((\sqrt a)^{2}=a\).
- Only like surds add or subtract.
- Expand like algebra; answer as \(a+b\sqrt c\).
If you see …
- √ of a non-squarelook for a square factor.
- surds with + or −simplify each first.
- "in the form a + b√c"expand and collect.
Practice
Practice: easy, medium, hard
Easy
Show working
- \(\sqrt{100\times3}\)
Answer \(10\sqrt3\)
Show working
- \(\sqrt{36}\)
Answer 6
Show working
- \(2\sqrt3+5\sqrt3\)
Answer \(7\sqrt3\)
Medium
Guide me
What is it asking? one surd
What do we know? 200 and 32
Plan square factors 100 and 16
Show working
- \(10\sqrt2+4\sqrt2\)
Answer \(14\sqrt2\)
Guide me
What is it asking? a + b√3
What do we know? a bracket squared
Plan three terms
Show working
- \(25-10\sqrt3+3\)
Answer \(28-10\sqrt3\)
Guide me
What is it asking? a + b√5
What do we know? two brackets
Plan four products
Show working
- \(3-\sqrt5+3\sqrt5-5\)
Answer \(-2+2\sqrt5\)
Hard
Guide me
What is it asking? a + b√2
What do we know? four products
Plan \(\sqrt2\times4\sqrt2=8\)
Show working
- \(3+12\sqrt2+\sqrt2+8\)
Answer \(11+13\sqrt2\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| four terms | M1 | ||
| \(11+13\sqrt2\) | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? k
What do we know? \(\sqrt{320}\)
Plan square factor 64
Show working
- \(\sqrt{64\times5}=8\sqrt5\)
Answer 8
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\sqrt{320}\) | M1 | ||
| 8 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? simplest form
What do we know? multiply each term by √3
Plan \(\sqrt{18}=3\sqrt2\)
Show working
- \(6+\sqrt{18}\)
Answer \(6+3\sqrt2\)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(6+\sqrt{18}\) | M1 | ||
| \(6+3\sqrt2\) | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can write a surd in its simplest form.
- I can add and subtract surds after simplifying.
- I can multiply surds and expand brackets into a + b√c.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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