Year 10 Mathematics Core — Surds

Cambridge Ch 4: Sections 4A & 4B  •  Wed 27 May 2026
📚 Continue with: → Ex 4C — Multiplying & Dividing → Ex 4D — Rationalising the Denominator

Today's lesson

We're starting Chapter 4 — Surds. By the end of today you should be able to:

Learning intentions

Part 1 — Rational vs irrational (Warm up, ~5 min)

Every real number sits somewhere on the number line. We split real numbers into two families:

Rational numbers
Can be written as a fraction $\dfrac{a}{b}$.
Their decimals terminate or recur.
e.g. $\tfrac{1}{2}=0.5$,   $\tfrac{1}{3}=0.\overline{3}$,   $7$,   $-1.6$,   $\sqrt{9}=3$
Irrational numbers
Cannot be written as a fraction.
Their decimals never end and never repeat.
e.g. $\pi$,   $\sqrt{2}$,   $\sqrt{7}$,   $2\sqrt{3}-1$
A surd is an irrational number written with a root sign, like $\sqrt{2}, \sqrt{15}, 1+\sqrt{5}$.
$\sqrt{9}$ is not a surd because $\sqrt{9}=3$ (it simplifies to a rational number).
EXAMPLE 1 — Surd or not?
Decide which of the following are surds:   $\sqrt{16}, \;\; \sqrt{20}, \;\; \sqrt{49}, \;\; \sqrt{31}, \;\; 2\sqrt{3}, \;\; \pi$

Working:

  • $\sqrt{16}=4$ → rational, not a surd
  • $\sqrt{20}$ → irrational, surd
  • $\sqrt{49}=7$ → rational, not a surd
  • $\sqrt{31}$ → irrational, surd
  • $2\sqrt{3}$ → irrational with a root sign, surd
  • $\pi$ → irrational but no root sign — irrational, not a surd

Trick: a square root is not a surd when the number under it is a perfect square (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …)

Practice 1.1 — Sort these into SURD or NOT A SURD:

$\sqrt{25}, \;\; \sqrt{30}, \;\; \sqrt{81}, \;\; \sqrt{12}, \;\; \sqrt{100}, \;\; \sqrt{2}, \;\; \sqrt{75}, \;\; \sqrt{1}$

Surds: $\sqrt{30}, \sqrt{12}, \sqrt{2}, \sqrt{75}$ (the number under the root is not a perfect square)
Not surds: $\sqrt{25}=5, \sqrt{81}=9, \sqrt{100}=10, \sqrt{1}=1$

Part 2 — Simplifying surds (4A, ~15 min)

The rule

For $a, b \ge 0$:

$\sqrt{a \times b} \;=\; \sqrt{a} \times \sqrt{b}$

To simplify a surd, find the largest perfect square factor of the number under the root, then split it.

📺 Walkthrough: simplifying $\sqrt{50}$ by pulling out the largest perfect-square factor.

EXAMPLE 2
Simplify $\sqrt{50}$.
  1. Largest square factor of 50?   $50 = 25 \times 2$   (25 is a perfect square)
  2. Split it up: $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25}\times\sqrt{2}$
  3. Simplify: $= 5\sqrt{2}$

Check: $50 = 25\times 2$ ✓   and 25 is the biggest square that divides 50.

EXAMPLE 3
Simplify $\sqrt{72}$.

$72 = 36 \times 2$  →  $\sqrt{72} = \sqrt{36}\times\sqrt{2} = 6\sqrt{2}$

What if you didn't spot 36? You might write $72 = 9 \times 8$, getting $\sqrt{72} = 3\sqrt{8}$. That's not fully simplified! You'd then need to simplify $\sqrt{8} = 2\sqrt{2}$, giving $3 \times 2\sqrt{2} = 6\sqrt{2}$. Same answer — just more work. Try to spot the biggest square factor first.

EXAMPLE 4
Simplify $3\sqrt{48}$.
  1. Simplify the surd part first: $48 = 16\times 3$
  2. $\sqrt{48} = \sqrt{16}\times\sqrt{3} = 4\sqrt{3}$
  3. Multiply by the outside number: $3\sqrt{48} = 3 \times 4\sqrt{3} = 12\sqrt{3}$

Practice 2.1 — Simplify:

  1. $\sqrt{8}$
  2. $\sqrt{18}$
  3. $\sqrt{27}$
  4. $\sqrt{45}$
  5. $\sqrt{75}$
  6. $\sqrt{200}$
a) $\sqrt{8}=\sqrt{4\times 2}=2\sqrt{2}$
b) $\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}$
c) $\sqrt{27}=\sqrt{9\times 3}=3\sqrt{3}$
d) $\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}$
e) $\sqrt{75}=\sqrt{25\times 3}=5\sqrt{3}$
f) $\sqrt{200}=\sqrt{100\times 2}=10\sqrt{2}$

Practice 2.2 — Simplify (numbers in front):

  1. $2\sqrt{12}$
  2. $5\sqrt{20}$
  3. $3\sqrt{32}$
  4. $4\sqrt{98}$
a) $2\sqrt{12}=2\sqrt{4\times 3}=2\times 2\sqrt{3}=4\sqrt{3}$
b) $5\sqrt{20}=5\sqrt{4\times 5}=5\times 2\sqrt{5}=10\sqrt{5}$
c) $3\sqrt{32}=3\sqrt{16\times 2}=3\times 4\sqrt{2}=12\sqrt{2}$
d) $4\sqrt{98}=4\sqrt{49\times 2}=4\times 7\sqrt{2}=28\sqrt{2}$
EXAMPLE 5
Express $5\sqrt{3}$ as a single square root.

This is the reverse of simplifying! Bring the 5 inside the root by squaring it:

$5\sqrt{3} = \sqrt{5^2 \times 3} = \sqrt{25 \times 3} = \sqrt{75}$

Practice 2.3 — Express as a single square root:

  1. $2\sqrt{5}$
  2. $3\sqrt{2}$
  3. $4\sqrt{7}$
  4. $6\sqrt{3}$
a) $2\sqrt{5} = \sqrt{4\times 5} = \sqrt{20}$
b) $3\sqrt{2} = \sqrt{9\times 2} = \sqrt{18}$
c) $4\sqrt{7} = \sqrt{16\times 7} = \sqrt{112}$
d) $6\sqrt{3} = \sqrt{36\times 3} = \sqrt{108}$

Part 3 — Adding and subtracting surds (4B, ~15 min)

The rule

You can only add or subtract like surds — surds with the same number under the root.

$a\sqrt{x} + b\sqrt{x} = (a+b)\sqrt{x}$

Treat the surd like a pronumeral: $3\sqrt{2} + 5\sqrt{2}$ behaves exactly like $3x + 5x = 8x$.

⚠️ You CANNOT do $\sqrt{2}+\sqrt{3} = \sqrt{5}$. That is a common error — the numbers under the root must be the same to combine them.
EXAMPLE 6 — Easy like surds
Simplify each:
(a) $3\sqrt{5} + 7\sqrt{5}$   (b) $9\sqrt{2} - 4\sqrt{2}$   (c) $4\sqrt{3} + 2\sqrt{5} - \sqrt{3} + 6\sqrt{5}$

(a) $3\sqrt{5} + 7\sqrt{5} = 10\sqrt{5}$

(b) $9\sqrt{2} - 4\sqrt{2} = 5\sqrt{2}$

(c) $4\sqrt{3} + 2\sqrt{5} - \sqrt{3} + 6\sqrt{5} \;=\; (4-1)\sqrt{3} + (2+6)\sqrt{5} \;=\; 3\sqrt{3}+8\sqrt{5}$

📺 Walkthrough: $\sqrt{18}+\sqrt{50}$ — simplify each surd first, then combine like terms.

EXAMPLE 7 — Simplify first, THEN combine
Simplify $\sqrt{18} + \sqrt{50}$.
  1. These don't look like like surds — but simplify each:
  2. $\sqrt{18}=\sqrt{9\times 2}=3\sqrt{2}$
  3. $\sqrt{50}=\sqrt{25\times 2}=5\sqrt{2}$
  4. Now they ARE like surds: $3\sqrt{2}+5\sqrt{2}=8\sqrt{2}$

Golden rule: always simplify each surd first. They might secretly be like surds.

EXAMPLE 8 — Mixed
Simplify $\sqrt{48} - \sqrt{75} + \sqrt{12}$.
  • $\sqrt{48}=\sqrt{16\times 3}=4\sqrt{3}$
  • $\sqrt{75}=\sqrt{25\times 3}=5\sqrt{3}$
  • $\sqrt{12}=\sqrt{4\times 3}=2\sqrt{3}$

$= 4\sqrt{3} - 5\sqrt{3} + 2\sqrt{3} = (4-5+2)\sqrt{3} = \sqrt{3}$

Practice 3.1 — Add or subtract:

  1. $4\sqrt{3} + 5\sqrt{3}$
  2. $8\sqrt{7} - 3\sqrt{7}$
  3. $6\sqrt{2} + \sqrt{2} - 4\sqrt{2}$
  4. $5\sqrt{6} + 3\sqrt{2} - 2\sqrt{6} + \sqrt{2}$
a) $9\sqrt{3}$
b) $5\sqrt{7}$
c) $(6+1-4)\sqrt{2} = 3\sqrt{2}$
d) $(5-2)\sqrt{6} + (3+1)\sqrt{2} = 3\sqrt{6}+4\sqrt{2}$

Practice 3.2 — Simplify first, then combine:

  1. $\sqrt{8}+\sqrt{32}$
  2. $\sqrt{27}-\sqrt{12}$
  3. $\sqrt{45}+\sqrt{20}$
  4. $\sqrt{72}-\sqrt{50}+\sqrt{18}$
  5. $2\sqrt{48}+3\sqrt{27}$
a) $2\sqrt{2}+4\sqrt{2}=6\sqrt{2}$
b) $3\sqrt{3}-2\sqrt{3}=\sqrt{3}$
c) $3\sqrt{5}+2\sqrt{5}=5\sqrt{5}$
d) $6\sqrt{2}-5\sqrt{2}+3\sqrt{2}=4\sqrt{2}$
e) $2(4\sqrt{3})+3(3\sqrt{3}) = 8\sqrt{3}+9\sqrt{3}=17\sqrt{3}$

Part 4 — Quick quiz (5 min, mark yourself)

Pick the correct answer for each. Click Mark at the bottom.

Q1. Which of these is a surd?

Q2. Simplify $\sqrt{28}$.

Q3. Simplify $\sqrt{98}$.

Q4. Simplify $3\sqrt{5}+7\sqrt{5}-2\sqrt{5}$.

Q5. Simplify $\sqrt{50}+\sqrt{8}$.

Q6. Express $4\sqrt{3}$ as a single square root.

Working program — Cambridge textbook

After you've finished the quiz, open Cambridge Chapter 4 and complete these questions in your exercise book:

ExerciseStandard pathway (set work)
4A — Irrational numbers & surdsQ11, Q12
4B — Adding/subtracting surdsQ7, Q8*, Q10*

* = harder, attempt if time. If you finish early, start 4C (Multiplying & Dividing Surds).

Exit ticket — write in your book

Before you pack up, in your exercise book write one sentence each:
  1. What is a surd?
  2. What is the first step when simplifying a surd like $\sqrt{72}$?
  3. Why can't you add $\sqrt{2}+\sqrt{3}$ together?
I'll check exit tickets when I'm back. Thanks team 🙏 — Mr Wong