Core 4.2 Surds

📋 Surd Rules — Quick Reference

1
√(a × b) = √a × √b — split a surd into factors
2
√a × √a = a — squaring removes the root entirely
3
Add/subtract surds: like terms only — k√n ± m√n = (k±m)√n
4
Simplify: find the largest perfect square factor, extract it
5
Rationalise: multiply top & bottom by the surd denominator
6
Conjugate: (a+√b)(a−√b) = a² − b — eliminates the surd
🧠

Key Words — What Do They Mean?

SurdAn irrational root
e.g. √2, √3, √5
IrrationalCan't be written
as a fraction
RadicandThe number
inside √
SimplifyExtract perfect
square factors
RationaliseRemove surds
from denominator
ConjugateFlip the sign:
a+√b → a−√b
✂️
Simplify √n
√
Press Go
➕
Multiply Surds
√ × √
Press Go
🔄
Rationalise 1/√n
1 / √
Press Go

Surds

A surd is an irrational root that can't be written as a fraction — like √2 or √3. Engineers use exact surd form to keep calculations precise before rounding at the final step.

Exact values Simplifying surds Rationalising denominators Expanding brackets Engineering precision
What is a Surd?

A surd is a root that gives an irrational number — it can't be expressed as a fraction and its decimal never terminates or repeats.

√2 = 1.41421356… (never ends)
√4 = 2 ✅ NOT a surd (exact)
√9 = 3 ✅ NOT a surd (exact)
💡 Perfect squares (1, 4, 9, 16, 25…) are NOT surds — they have exact integer roots.
Why Use Surds?
  1. Exact — √3 is more precise than 1.732 (which is rounded)
  2. Simpler — 2√2 is cleaner than 2.828427…
  3. Required — exam questions often say "leave in surd form"
  4. Engineering — diagonal calculations, AC circuits and stress analysis all use surds
√
What is a Surd?
recognise & identify →
✂️
Simplifying
√12 → 2√3 →
➕
Operations
add, subtract, multiply →
📐
Expanding Brackets
(a + √b)² →
🔄
Rationalising
remove √ from denominator →
⚙️
Engineering Uses
where you'll need this →
√

What is a Surd? — At a Glance

📍 Three things to check:
1
Is it a root (√, ∛ etc.)? → could be a surd
2
Is the answer a whole number? (√4=2, √9=3) → NOT a surd
3
Is the decimal endless and random? (√2=1.41421…) → IS a surd ✓
💡
Memory trick: Perfect squares (1,4,9,16,25,36…) are never surds
√

What is a Surd?

Learn to identify surds and understand the difference between rational and irrational roots.

IrrationalPerfect squares√ notation
🔍 Surd Checker — is √n a surd?
📍 Surds on the number line — drag the slider to explore
√2
📌
Key Rule
√n is a surd if n is NOT a perfect square (1, 4, 9, 16, 25, 36, 49…)
Surds ✅
√2 ≈ 1.41421… (irrational)
√3 ≈ 1.73205… (irrational)
√5 ≈ 2.23606… (irrational)
√7 ≈ 2.64575… (irrational)
√50 ≈ 7.07106… (irrational)
✅ These cannot be written exactly as fractions or terminating decimals.
NOT Surds ❌
√1 = 1    (1 = 1²)
√4 = 2    (4 = 2²)
√9 = 3    (9 = 3²)
√16 = 4   (16 = 4²)
√25 = 5   (25 = 5²)
❌ These are perfect squares — their roots are whole numbers, so they are NOT surds.
Watch out!Cube & higher roots

Surds aren't just square roots. ∛2 and ⁴√5 are also surds — any root that's irrational counts.

∛8 = 2 ✅ NOT a surd (2³ = 8)
∛2 ≈ 1.2599… → IS a surd
Key Property√a × √a = a

The most important surd identity — squaring a surd removes the root entirely.

(√3)² = √3 × √3 = 3
(√7)² = 7
(√n)² = n
This is the key to rationalising denominators later!
✂️

Simplifying Surds — Step by Step

Follow every time:
1
List factors of n — which ones are perfect squares? (4, 9, 16, 25, 36…)
2
Pick the largest perfect square factor
3
Split: √(square × rest) = √square × √rest
4
Evaluate √square → gives a whole number coefficient
✅
Result: k√m where m has no perfect square factors left
✂️

Simplifying Surds

Write √n in its simplest form by extracting the largest perfect square factor.

Perfect square factors√(ab) = √a × √bSimplest form
✂️ Surd Simplifier — enter any √n
📐
The Rule
√(a × b) = √a × √b  —  extract the largest perfect square factor
MethodStep by step
  1. Find the largest perfect square that divides into n
  2. Write n = perfect square × remainder
  3. Split: √n = √(square) × √(remainder)
  4. Evaluate the perfect square root: √4=2, √9=3, √16=4…
  5. Write the result: k√m where m has no perfect square factors
Worked ExampleSimplify √72
Factors of 72: 4×18, 9×8, 36×2
Largest perfect square factor = 36
√72 = √(36 × 2)
= √36 × √2
= 6√2 ✅
✅ Always use the LARGEST perfect square factor — otherwise you might need to simplify again.
✏️ Worked step-through — Simplify √48
Start:Simplify √48 — find the largest perfect square factor of 48
Factors:48 = 4 × 12  |  48 = 16 × 3  ← 16 is larger perfect square
Split:√48 = √(16 × 3) = √16 × √3
Evaluate:√16 = 4
Answer:4√3 ✅   (check: 3 has no perfect square factors)
Step 1 of 5
Common simplifications to know
√8 = 2√2
√12 = 2√3
√18 = 3√2
√20 = 2√5
√27 = 3√3
√32 = 4√2
√45 = 3√5
√50 = 5√2
√75 = 5√3

📋 Rules: Adding, Subtracting & Multiplying Surds

➕
Add/subtract like surds only: 3√2 + 5√2 = 8√2 — the √ must match
✖
Multiply: a√m × b√n = ab√(mn) — multiply coefficients & radicands separately
💡
√a × √a = a — the most important surd identity
⚠️
√2 + √3 ≠ √5 — you cannot add under the root!
➕

Operations with Surds

Add, subtract and multiply surds — the rules are similar to collecting like terms in algebra.

Like surdsCollecting terms√a × √b = √(ab)
🧮 Surd Operations Calculator
√
√
Adding & SubtractingOnly combine LIKE surds

Like surds have the same radicand (number under the √). Treat them like algebra — combine coefficients only.

3√2 + 5√2 = 8√2 ✅
7√3 − 2√3 = 5√3 ✅
❌ 3√2 + 4√3 → cannot combine (different surds)
💡 Think of it like algebra: 3x + 5x = 8x, so 3√2 + 5√2 = 8√2
MultiplyingMultiply coefficients & radicands separately
a√b × c√d = (ac)√(bd)
3√2 × 4√3 = 12√6
2√5 × √5 = 2 × 5 = 10
√6 × √6 = 6
Key: √n × √n = n — squaring removes the surd!
✏️ Step-through — Simplify 3√12 + 2√27
Start:3√12 + 2√27 — surds look different, but they might be like terms after simplifying
Simplify √12:√12 = √(4×3) = 2√3  →  3√12 = 3 × 2√3 = 6√3
Simplify √27:√27 = √(9×3) = 3√3  →  2√27 = 2 × 3√3 = 6√3
Combine:6√3 + 6√3 = 12√3 ✅
Step 1 of 4
📐

Expanding Brackets with Surds

Same as normal algebra — just extra surd rules:
F
First terms: multiply first terms together
O
Outer terms: first × last
I
Inner terms: last of first bracket × first of second
L
Last terms: remember √n × √n = n
💡
Key: (a+√b)(a−√b) = a²−b — the conjugate pair eliminates surds!
📐

Expanding Brackets with Surds

Apply FOIL and standard expansion — surds behave exactly like algebra variables.

FOIL(a+√b)²Difference of two squares
📐 Bracket Expander — (a + b√c)(d + e√f)
+ √
×
+ √
Single bracketDistribute across surds
√2(3 + √2) = 3√2 + (√2)² = 3√2 + 2
√3(√12 − 1) = √36 − √3 = 6 − √3
💡 Use √a × √b = √(ab), then simplify the result.
Double brackets — FOIL(a + √b)(c + √d)
(2 + √3)(1 + √3)
= 2×1 + 2×√3 + √3×1 + √3×√3
= 2 + 2√3 + √3 + 3
= 5 + 3√3 ✅
Perfect square(a + √b)² formula
(a + √b)² = a² + 2a√b + b
(3 + √2)² = 9 + 6√2 + 2 = 11 + 6√2
(1 − √5)² = 1 − 2√5 + 5 = 6 − 2√5
Difference of two squares(a + √b)(a − √b)
(a + √b)(a − √b) = a² − b
(3 + √2)(3 − √2) = 9 − 2 = 7
(√5 + 1)(√5 − 1) = 5 − 1 = 4
✅ The result is always a rational number — no surds! This is the key to rationalising.
🔄

Rationalising Denominators — Step by Step

Two cases to know:
A
Simple surd (1/√n): multiply top & bottom by √n → get √n/n
B
Conjugate (a±√b): multiply by the conjugate (a∓√b) to clear the surd
💡
Why? (a+√b)(a−√b) = a²−b — no surd in denominator!
✅
Simplify the result — check if numerator & denominator share a factor
🔄

Rationalising the Denominator

Remove surds from the bottom of a fraction — examiners always expect this. It's the polished final form.

ConjugateMultiply top & bottomRational denominator
🔄 Rationalise the Denominator
Case 1 — Simple surd1 / √a

Multiply top AND bottom by √a. The denominator becomes (√a)² = a — rational!

1/√3 × √3/√3 = √3/3 ✅
5/√2 × √2/√2 = 5√2/2 ✅
6/√12 = 6/2√3 = 3/√3 = 3√3/3 = √3 ✅
✅ Always simplify the surd first, then rationalise.
Case 2 — Binomial surd1 / (a ± √b)

Multiply top AND bottom by the conjugate — flip the sign of the surd part. Uses difference of two squares.

1/(2+√3) × (2−√3)/(2−√3)
= (2−√3)/(4−3)
= (2−√3)/1
= 2−√3 ✅
Conjugate of (a + √b) is (a − √b) — just flip the sign!
✏️ Step-through — Rationalise 6 / (3 + √3)
Start:6 / (3 + √3) — conjugate of (3 + √3) is (3 − √3)
Multiply:= [6 × (3 − √3)] / [(3 + √3)(3 − √3)]
Denominator:(3 + √3)(3 − √3) = 3² − (√3)² = 9 − 3 = 6
Numerator:6 × (3 − √3) = 18 − 6√3
Divide:(18 − 6√3) / 6 = 3 − √3 ✅
Step 1 of 5

⚙️ Surds in Engineering — Key Applications

📐
Diagonal: square with side a → diagonal = a√2 (Pythagoras)
⚡
AC circuits: impedance Z = √(R²+X²) — often a surd exact value
🔩
Stress analysis: shear stress uses √3 — keep exact, round at end
💡
Always: work with exact surds through the problem — round only at the final step
⚙️ Diagonal Calculator — a² + b² = c²
⚙️

Surds in Engineering

Where exact surd values appear in real engineering problems — precision matters.

Structures
🔩 Diagonal Lengths — Pythagoras
A square steel plate has side 5 m. The diagonal = √(5² + 5²) = √50 = 5√2 m ≈ 7.071 m. Working in exact form 5√2 avoids rounding error in further calculations.
diagonal = √(a² + b²)
Square: √(5²+5²) = √50 = 5√2 m
Rectangle 3×4: √(9+16) = √25 = 5 m ✅
Electrical
⚡ AC Circuits — RMS Values
The RMS value of a sinusoidal AC signal is V_peak / √2. For a 230 V RMS supply, V_peak = 230√2 ≈ 325.27 V. Engineers write 230√2 exactly.
V_rms = V_peak / √2
V_peak = V_rms × √2 = 230√2 V
Rationalised: 230√2/2 → 115√2 (half-wave)
Stress Analysis
📐 Shear Stress — von Mises
The shear yield stress τ = σ_y / √3. If σ_y = 250 MPa, then τ = 250/√3 = 250√3/3 ≈ 144.3 MPa (rationalised form).
τ = σ_y / √3 = σ_y × √3 / 3
τ = 250/√3 = 250√3/3 MPa
Always rationalise before final answer!
Vibration
🔧 Natural Frequency
ωn = √(k/m) for a spring-mass system. If k = 400 N/m, m = 4 kg: ωn = √100 = 10 rad/s. Often results in surds when k/m is not a perfect square.
ωn = √(k/m)
k=200, m=3: ωn = √(200/3) = 10√6/3 rad/s
🎮

Quick Fire Quiz

10 questions — instant feedback. Test your surd skills!

⚡ Surds Blitz

Score: 0 Streak: 0 🔥 Q 1/10
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📝

Practice Questions

15 questions in 3 tiers. Type your answer then reveal full working.

🟢 Tier 1 — Foundations
Q1
Is √49 a surd? Explain why or why not.
[1]
No — 49 = 7², so √49 = 7 exactly.
NOT a surd ✅
Q2
Simplify √50
[2]
50 = 25 × 2   HCF perfect square = 25
√50 = √25 × √2 = 5√2 ✅
Q3
Simplify √75
[2]
75 = 25 × 3
√75 = √25 × √3 = 5√3 ✅
Q4
Calculate 3√5 × 2√5
[2]
3 × 2 = 6   √5 × √5 = 5
= 6 × 5 = 30 ✅
Q5
Add: 3√2 + 7√2
[1]
Like surds — add coefficients: 3 + 7 = 10
10√2 ✅
🟡 Tier 2 — Core Skills
Q6
Simplify √8 + √18
[3]
√8 = 2√2   √18 = 3√2
2√2 + 3√2 = 5√2 ✅
Q7
Expand and simplify: (1 + √3)²
[3]
(1+√3)² = 1 + 2√3 + (√3)² = 1 + 2√3 + 3 = 4 + 2√3 ✅
Q8
Expand: (2 + √5)(2 − √5)
[2]
Difference of two squares: 2² − (√5)² = 4 − 5 = −1 ✅
Q9
Rationalise the denominator: 3 / √3
[2]
3/√3 × √3/√3 = 3√3/3 = √3 ✅
Q10
Rationalise: 10 / (3 + √2)
[3]
× (3−√2)/(3−√2): numerator = 30−10√2
denominator = 9−2 = 7
= (30−10√2)/7 ✅
🔴 Tier 3 — Challenge
Q11
Simplify: 3√12 − √75 + 2√27
[4]
3√12=6√3   √75=5√3   2√27=6√3
6√3−5√3+6√3 = 7√3 ✅
Q12
Expand and simplify: (3 + 2√2)(1 + √2)
[3]
3+3√2+2√2+2(2) = 3+5√2+4 = 7+5√2 ✅
Q13
⚙️ A square cross-section strut has area 45 cm². Find the exact side length.
[3]
Side = √45 = √(9×5) = 3√5 cm ✅
Q14
⚡ Rationalise 230/√2 to show V_rms from V_peak.
[3]
230/√2 × √2/√2 = 230√2/2 = 115√2 V ✅
Q15
Show (√6+√2)(√6−√2)=4, hence find 1/(√6−√2) rationalised.
[4]
(√6+√2)(√6−√2)=6−2=4 ✅
1/(√6−√2)×(√6+√2)/(√6+√2) = (√6+√2)/4 ✅
🎯 Score guide: Q1–5 foundations (8 marks) · Q6–10 core (13 marks) · Q11–15 challenge (17 marks)
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