Surds and Operations
[PATHS extension] Surds are irrational numbers that cannot be simplified to neat decimals. Learn to tame them — simplify, combine and rationalise.
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Worksheet
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Q1: A calculator gives $\sqrt{2} \approx 1.41421356...$ Why do mathematicians prefer to write $\sqrt{2}$ instead of its decimal approximation? Can you think of a situation where using the exact value matters?
Q2: Simplify $\sqrt{50}$ in your head as much as you can. What perfect square factor can you extract? What is the simplified form?
Learning Intentions
Know
- What a surd is and why it is irrational.
- The product rule: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$.
Understand
- Why exact surd form is preferred over decimal approximations.
- How to identify and combine like surds.
Can Do
- Simplify surds by extracting perfect square factors.
- Add, subtract, multiply and divide simple surds.
- Rationalise simple denominators containing surds.
Success Criteria
- I can simplify surds such as $\sqrt{50}$ to $5\sqrt{2}$.
- I can add and subtract like surds: $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$.
- I can multiply surds: $\sqrt{3} \times \sqrt{6} = \sqrt{18} = 3\sqrt{2}$.
- I can rationalise simple denominators: $\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$.
Key Terms
Common Mistakes to Avoid
Wrong: “$\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$”. The square root of a sum is NOT the sum of the square roots.
Right: $\sqrt{9 + 16} = \sqrt{25} = 5$. You must add first, then take the root.
Wrong: “$\sqrt{3} + \sqrt{5} = \sqrt{8}$”. Unlike surds cannot be combined. $\sqrt{3} + \sqrt{5}$ stays as it is.
Right: Only like surds can be added or subtracted: $4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}$.
A surd is in simplest form when the number under the root has no perfect square factors (other than 1).
To simplify, factor out the largest perfect square:
$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$
Perfect squares to know: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Always look for the largest perfect square factor.
What to write in your book
- A surd is in simplest form when the number under the root has no perfect square factors
- Factor out the largest perfect square: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$
- Product rule: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
- Know your perfect squares: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Correct! $\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$.
Not quite. $\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}$.
Mathematicians prefer denominators without surds. Rationalising removes the surd from the bottom of a fraction.
To rationalise $\dfrac{1}{\sqrt{a}}$, multiply top and bottom by $\sqrt{a}$:
$\dfrac{1}{\sqrt{2}} = \dfrac{1}{\sqrt{2}} \times \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$
This works because $\sqrt{a} \times \sqrt{a} = a$, which is rational.
What to write in your book
- To rationalise $\frac{1}{\sqrt{a}}$, multiply top and bottom by $\sqrt{a}$
- Result: $\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$
- This works because $\sqrt{a} \times \sqrt{a} = a$, which is rational
Correct! $\frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5}$.
Not quite. $\frac{5}{\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{5} = \sqrt{5}$.
Interactive: Surd Simplifier
Your Turn
Question 1: Simplify $\sqrt{48}$.
Question 2: Simplify $3\sqrt{2} + 5\sqrt{2} - 2\sqrt{2}$.
Question 3: Rationalise $\dfrac{2}{\sqrt{3}}$.
Revisit Your Thinking
Look back at your Think First answer about why mathematicians prefer $\sqrt{2}$ over $1.414...$. Calculate $(1.414)^2$ and compare it to 2. What does this tell you about the importance of exact values? Explain why engineers might use surds in calculations before converting to decimals at the final step.
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
Simplify $\sqrt{75}$.
Simplify $4\sqrt{7} + 3\sqrt{7}$.
Rationalise $\dfrac{5}{\sqrt{5}}$.
Simplify $\sqrt{8} \times \sqrt{2}$.
Which of the following is NOT equal to $\sqrt{18}$?
Simplify the following surds.
(a) $\sqrt{200}$ (1 mark)
(b) $\sqrt{98}$ (1 mark)
(c) $2\sqrt{12} + 3\sqrt{3}$ (1 mark)
(a) Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$. (2 marks)
(b) Rationalise $\dfrac{4}{\sqrt{8}}$, giving your answer in simplest form. (2 marks)
A square has area 50 cm$^2$.
(a) Find the exact side length of the square, in simplest surd form. (2 marks)
(b) A student says the side length is approximately 7.07 cm. Explain why the exact answer $\sqrt{50}$ cm is more useful for further calculations. (2 marks)
(c) Simplify $\sqrt{50}$. (1 mark)
Surd
Irrational root
Simplify
Extract perfect squares
Product rule
$\sqrt{ab} = \sqrt{a}\sqrt{b}$
Like surds
Same root part
Rationalise
Multiply by $\sqrt{a}/\sqrt{a}$
Exact value
Keep surd form
Real-Life Link
Surds appear throughout engineering and physics. When calculating the diagonal of a square with side length 1, the exact answer is $\sqrt{2}$ — an irrational number that appears in the design of every square or rectangular structure. In electrical engineering, the impedance of AC circuits involves $\sqrt{2}$ when converting between peak and RMS voltages. Architects use surds when working with $30\degree$-$60\degree$-$90\degree$ triangles, where side ratios are $1 : \sqrt{3} : 2$. Keeping answers in surd form until the final step prevents compounding rounding errors that could make buildings unsafe or circuits inefficient.
Game Time!
Test your surd skills in an interactive challenge.
Play Surds Challenge