Surds Explained Step by Step

🎯 By the end of this lesson...

You should be able to:

Explain what a surd is.
Distinguish surds from ordinary whole-number roots.
Recognise perfect squares.
Identify whether a square root is a surd.
Find the largest perfect-square factor of a number.
Simplify surds into exact form.
Add and subtract like surds.
Multiply simple surds.
Rationalise simple denominators.
Apply surds to practical and geometrical problems.

🤔 Think About This...

🌿

A square garden

A family wants to create a square garden with an area of 50 m².

Because the garden is square:

Area = side × side

Therefore, one side must have length:

√50 metres

But √50 is not a whole number.

Can it be written in a simpler exact form?

Surds allow us to keep square-root answers exact instead of replacing them with rounded decimal approximations.

1️⃣ Revisiting Square Roots

In the previous lesson, we learnt that a square root asks:

What number multiplied by itself gives the number inside the root?

Example 1: Find √25

5 × 5 = 25

√25 = 5

Example 2: Find √81

9 × 9 = 81

√81 = 9

Numbers such as 25 and 81 are called perfect squares.

2️⃣ Perfect Squares

A perfect square is a number obtained by multiplying a whole number by itself.

1² 1

2² 4

3² 9

4² 16

5² 25

6² 36

7² 49

8² 64

9² 81

10² 100

11² 121

12² 144

Recognising perfect squares quickly makes simplifying surds much easier.

3️⃣ What Is a Surd?

A surd is a root that cannot be simplified to a whole number or written exactly as a fraction of two integers.

✅ Surds

√2

√3

√5

√7

❌ Not Surds

√25 = 5

√64 = 8

√121 = 11

∛27 = 3

Not every square root is a surd. Always simplify the root before classifying it.

🔍 Exact and Approximate Values

Consider √2.

Exact value

√2

Exact

Approximate value

√2 ≈ 1.414

Approximate

The decimal value continues forever without repeating:

√2 = 1.414213562...

Writing √2 keeps the exact value. Writing 1.414 gives only an approximation.

💡 Did You Know?

Engineers, architects and scientists often keep answers in surd form until the final stage of a calculation.

This prevents rounding errors from building up.

Exact

5√2

Approximate

7.071...

Both values represent the same quantity, but the surd keeps the answer exact.

4️⃣ Is It a Surd?

Before deciding, simplify the root.

Square root

↓

Does it simplify to a whole number?

Yes

↓

Not a surd

No

↓

It may be a surd

Example 3: Is √49 a surd?

√49 = 7

√49 is not a surd because it simplifies to a whole number.

Example 4: Is √11 a surd?

11 is not a perfect square.

√11 cannot be written exactly as a whole number or fraction.

√11 is a surd.

🧠 StudyNest Method: Recognising a Surd

Step 1: Simplify the root if possible.

Step 2: Ask whether the number inside the root is a perfect square.

Step 3: If the root becomes a whole number, it is not a surd.

Step 4: If it cannot be written exactly as a whole number or fraction, it is a surd.

5️⃣ The Product Rule for Square Roots

To simplify surds, we use:

√(a × b) = √a × √b

This lets us split a square root into factors.

Example 5

√12

Since 12 = 4 × 3:

√12 = √(4 × 3)

= √4 × √3

= 2√3

We choose a perfect-square factor because its square root becomes a whole number.

6️⃣ Simplifying Surds

Simplifying a surd means removing every possible perfect-square factor from inside the root.

Example 6: Simplify √18

Find the largest perfect-square factor of 18.

18 = 1 × 18

18 = 9 × 2

√18 = √(9 × 2)

= √9 × √2

= 3√2

√18 simplifies to 3√2.

Example 7: Simplify √72

The largest perfect-square factor of 72 is 36.

72 = 36 × 2

√72 = √36 × √2

= 6√2

√72 simplifies to 6√2.

🧠 StudyNest Method: Simplifying a Surd

Step 1: Check whether the number inside the root is already a perfect square.

Step 2: If not, find its largest perfect-square factor.

Step 3: Write the number as:

perfect-square factor × remaining factor.

Step 4: Split the square root.

Step 5: Calculate the square root of the perfect-square factor.

Step 6: Check that the number remaining inside the root cannot be simplified further.

Look for the largest perfect-square factor first. This usually gives the shortest solution.

7️⃣ More Simplifying Examples

Example 8: Simplify √8

8 = 4 × 2

√8 = √4 × √2

√8 = 2√2

Example 9: Simplify √27

27 = 9 × 3

√27 = √9 × √3

√27 = 3√3

Example 10: Simplify √45

45 = 9 × 5

√45 = √9 × √5

√45 = 3√5

Example 11: Simplify √200

200 = 100 × 2

√200 = √100 × √2

√200 = 10√2

🌍 Real-Life Example: A Square Floor

Example 12

A square floor has an area of 72 m².

Find the exact length of one side.

Side length = √72

√72 = √(36 × 2)

= 6√2

The exact side length is 6√2 metres.

A decimal answer would be approximate. The surd form gives the exact length.

8️⃣ Simplifying Surds with Coefficients

A coefficient is the number written in front of a surd.

3√5

In this expression, 3 is the coefficient.

Example 13: Simplify 3√12

Simplify the surd first.

√12 = 2√3

Multiply the coefficient by 2.

3 × 2√3

3√12 = 6√3

Example 14: Simplify 5√20

√20 = √(4 × 5)

= 2√5

5 × 2√5 = 10√5

5√20 = 10√5

⚠ Common Simplifying Mistakes

Mistake 1:

Assuming every root is a surd.

√36 = 6, so it is not a surd.

Mistake 2:

Choosing a factor that does not help.

For √18, using 3 × 6 does not remove a square root. Using 9 × 2 does.

Mistake 3:

Stopping before the surd is fully simplified.

√48 = 4√3, not 2√12.

Mistake 4:

Splitting a sum incorrectly.

√(a + b) is generally not equal to √a + √b.

9️⃣ Like and Unlike Surds

Surds are called like surds when the root part is exactly the same.

✅ Like Surds

3√5 and 7√5

2√3 and −4√3

The number inside each square root is the same.

❌ Unlike Surds

3√2 and 3√5

√7 and 4√3

The root parts are different.

Like surds behave in a similar way to like terms in Algebra.

For example, 3x and 5x are like terms, while 3√2 and 5√2 are like surds.

🔟 Adding Like Surds

To add like surds, add their coefficients and keep the common surd.

Example 15: Simplify 3√5 + 2√5

Both terms contain √5, so they are like surds.

3√5 + 2√5

= (3 + 2)√5

3√5 + 2√5 = 5√5

Example 16: Simplify 4√7 + 9√7

4√7 + 9√7

= (4 + 9)√7

4√7 + 9√7 = 13√7

1️⃣1️⃣ Subtracting Like Surds

To subtract like surds, subtract their coefficients and keep the common surd.

Example 17: Simplify 9√3 − 4√3

9√3 − 4√3

= (9 − 4)√3

9√3 − 4√3 = 5√3

Example 18: Simplify 11√2 − 15√2

11√2 − 15√2

= (11 − 15)√2

11√2 − 15√2 = −4√2

⚠ Unlike Surds Cannot Be Collected

Example 19: Simplify 2√3 + 5√2

The root parts are different.

√3 ≠ √2

Therefore, the two terms cannot be collected.

The answer remains:

2√3 + 5√2

Do not add the numbers inside the roots.

√2 + √3 is not equal to √5.

🧠 StudyNest Method: Adding and Subtracting Surds

Step 1: Simplify every surd first.

Step 2: Check whether the root parts are identical.

Step 3: If they are identical, add or subtract the coefficients.

Step 4: Keep the common surd unchanged.

Step 5: If the root parts differ, leave the terms separate.

1️⃣2️⃣ Simplify Before Collecting

Surds that initially look different may become like surds after simplification.

Example 20: Simplify √12 + √27

Simplify each surd separately.

√12 = √(4 × 3) = 2√3

√27 = √(9 × 3) = 3√3

They are now like surds.

2√3 + 3√3

√12 + √27 = 5√3

Example 21: Simplify √48 + 2√27

√48 = √(16 × 3) = 4√3

2√27 = 2(3√3) = 6√3

4√3 + 6√3

√48 + 2√27 = 10√3

1️⃣3️⃣ Multiplying Simple Surds

When multiplying square roots, multiply the numbers inside the roots.

√a × √b = √(ab)

Example 22: Simplify √5 × √3

√5 × √3 = √15

√5 × √3 = √15

Example 23: Simplify √6 × √2

√6 × √2 = √12

Simplify √12.

√12 = √(4 × 3)

√6 × √2 = 2√3

1️⃣4️⃣ Multiplying Surds with Coefficients

Multiply the coefficients together and multiply the surds together.

Example 24: Simplify 2√3 × 4√5

Coefficients

2 × 4 = 8

Surds

√3 × √5 = √15

2√3 × 4√5 = 8√15

Example 25: Simplify 3√2 × 2√8

3 × 2 = 6

√2 × √8 = √16 = 4

6 × 4 = 24

3√2 × 2√8 = 24

1️⃣5️⃣ Squaring a Surd

A square root and a square undo each other.

(√a)² = a

Example 26: Calculate (√7)²

(√7)² = √7 × √7

= √49

(√7)² = 7

Example 27: Calculate (3√2)²

Square both the coefficient and the surd.

(3√2)² = 3² × (√2)²

= 9 × 2

(3√2)² = 18

1️⃣6️⃣ Rationalising a Simple Denominator

A rationalised denominator does not contain a surd.

Consider:

1/√2

Multiply the numerator and denominator by √2.

1/√2 × √2/√2

= √2/(√2 × √2)

= √2/2

The rationalised form of 1/√2 is √2/2.

Multiplying the numerator and denominator by the same non-zero value does not change the value of the fraction.

🧠 StudyNest Method: Rationalising a Denominator

Step 1: Identify the surd in the denominator.

Step 2: Multiply both the numerator and denominator by that surd.

Step 3: Multiply the numerator.

Step 4: Use √a × √a = a in the denominator.

Step 5: Simplify the final fraction if possible.

1️⃣7️⃣ More Rationalising Examples

Example 28: Rationalise 3/√5

3/√5 × √5/√5

= 3√5/5

3/√5 = 3√5/5

Example 29: Rationalise 5/√3

5/√3 × √3/√3

= 5√3/3

5/√3 = 5√3/3

Example 30: Rationalise 6/√2

6/√2 × √2/√2

= 6√2/2

Simplify by dividing by 2.

6/√2 = 3√2

🌍 Real-Life Example: Diagonal of a Square

Example 31

A square noticeboard has side length 5 metres.

The diagonal forms the hypotenuse of a right-angled triangle.

Diagonal² = 5² + 5²

= 25 + 25

= 50

Diagonal = √50

= √(25 × 2)

The exact diagonal is 5√2 metres.

🌍 Real-Life Example: Square Garden

Example 32

A square garden has an area of 200 m².

Find the exact side length.

Side = √200

= √(100 × 2)

= 10√2

Each side is exactly 10√2 metres.

🎯 Exam Tips

Memorise common perfect squares so you can recognise useful factors quickly.

Simplify every surd before trying to add or subtract terms.

If the question asks for an exact answer, do not replace the surd with a rounded decimal.

When rationalising a simple denominator, multiply the numerator and denominator by the surd in the denominator.

Check that no perfect-square factor remains inside the final root.

⚠ Common Mistakes

Mistake 1:

Thinking every square root is a surd.

√64 = 8, so it is not a surd.

Mistake 2:

Adding unlike surds.

√2 + √3 cannot be simplified to √5.

Mistake 3:

Collecting terms before simplifying them.

√12 and √27 become like surds only after simplification.

Mistake 4:

Forgetting to multiply the coefficients when multiplying surds.

Mistake 5:

Rationalising only the denominator.

You must multiply both numerator and denominator by the same surd.

Mistake 6:

Giving a decimal when the question requests an exact value.

🎮 Your Turn!

🟢 Foundation Questions

State whether √36 is a surd.
State whether √13 is a surd.
Simplify √8.
Simplify √18.
Simplify √27.
Simplify √45.
Simplify √75.
Simplify √200.

🔵 Developing Questions

Simplify 3√12.
Simplify 5√20.
Simplify 4√50.
Simplify 2√3 + 7√3.
Simplify 9√5 − 4√5.
Simplify √12 + √27.
Simplify √6 × √2.
Simplify 3√2 × 2√7.

🔴 Challenge Questions

Simplify √48 + 2√27.
Simplify 3√8 + √18.
Simplify (4√3)².
Rationalise 5/√3.
Rationalise 7/√2.
Rationalise 8/√5.
A square floor has an area of 98 m². Find the exact length of one side in simplest surd form.
A square garden has an area of 200 m². Find the exact side length.

✅ Show Answers

No. √36 = 6.
Yes. √13 cannot be written exactly as a whole number or fraction.
√8 = 2√2
√18 = 3√2
√27 = 3√3
√45 = 3√5
√75 = 5√3
√200 = 10√2
3√12 = 3(2√3) = 6√3
5√20 = 5(2√5) = 10√5
4√50 = 4(5√2) = 20√2
2√3 + 7√3 = 9√3
9√5 − 4√5 = 5√5
√12 + √27

= 2√3 + 3√3

= 5√3
√6 × √2 = √12 = 2√3
3√2 × 2√7 = 6√14
√48 + 2√27

= 4√3 + 6√3

= 10√3
3√8 + √18

= 3(2√2) + 3√2

= 6√2 + 3√2

= 9√2
(4√3)² = 16 × 3 = 48
5/√3 = 5√3/3
7/√2 = 7√2/2
8/√5 = 8√5/5
√98 = √(49 × 2) = 7√2 m
√200 = √(100 × 2) = 10√2 m

🧠 Remember This

A surd is an irrational root kept in exact form.

Not every root is a surd.

To simplify, find the largest perfect-square factor.

Only like surds can be added or subtracted.

Multiply coefficients and root parts separately.

Rationalising removes a surd from the denominator.

Keep a surd when the question asks for an exact answer.

🎯 Before Moving On...

✅ Can I explain what a surd is?
✅ Can I distinguish perfect-square roots from surds?
✅ Can I simplify a surd fully?
✅ Can I simplify a surd with a coefficient?
✅ Can I identify like and unlike surds?
✅ Can I add and subtract like surds?
✅ Can I multiply simple surds?
✅ Can I rationalise a simple denominator?
✅ Can I distinguish an exact value from an approximation?

🎉 Well Done!

You can now recognise, simplify and calculate with surds.

Remember:

Simplify first → Check for like surds → Use the correct operation → Keep the answer exact

📈 Your Progress

Lesson 10 of 17

Real Numbers Progress 59%

Surds

🎬 Keep exact values exact

🎯 By the end of this lesson...

🤔 Think About This...

A square garden

1️⃣ Revisiting Square Roots

Example 1: Find √25

Example 2: Find √81

2️⃣ Perfect Squares

3️⃣ What Is a Surd?

✅ Surds

❌ Not Surds

🔍 Exact and Approximate Values

Exact value

Approximate value

💡 Did You Know?

Exact

Approximate

4️⃣ Is It a Surd?

Example 3: Is √49 a surd?

Example 4: Is √11 a surd?

🧠 StudyNest Method: Recognising a Surd

5️⃣ The Product Rule for Square Roots

Example 5

6️⃣ Simplifying Surds

Example 6: Simplify √18

Example 7: Simplify √72

🧠 StudyNest Method: Simplifying a Surd

7️⃣ More Simplifying Examples

Example 8: Simplify √8

Example 9: Simplify √27

Example 10: Simplify √45

Example 11: Simplify √200

🌍 Real-Life Example: A Square Floor

Example 12

8️⃣ Simplifying Surds with Coefficients

Example 13: Simplify 3√12

Example 14: Simplify 5√20

⚠ Common Simplifying Mistakes

9️⃣ Like and Unlike Surds

✅ Like Surds

❌ Unlike Surds

🔟 Adding Like Surds

Example 15: Simplify 3√5 + 2√5

Example 16: Simplify 4√7 + 9√7

1️⃣1️⃣ Subtracting Like Surds

Example 17: Simplify 9√3 − 4√3

Example 18: Simplify 11√2 − 15√2

⚠ Unlike Surds Cannot Be Collected

Example 19: Simplify 2√3 + 5√2

🧠 StudyNest Method: Adding and Subtracting Surds

1️⃣2️⃣ Simplify Before Collecting

Example 20: Simplify √12 + √27

Example 21: Simplify √48 + 2√27

1️⃣3️⃣ Multiplying Simple Surds

Example 22: Simplify √5 × √3

Example 23: Simplify √6 × √2

1️⃣4️⃣ Multiplying Surds with Coefficients

Example 24: Simplify 2√3 × 4√5

Coefficients

Surds

Example 25: Simplify 3√2 × 2√8

1️⃣5️⃣ Squaring a Surd

Example 26: Calculate (√7)²

Example 27: Calculate (3√2)²

1️⃣6️⃣ Rationalising a Simple Denominator

🧠 StudyNest Method: Rationalising a Denominator

1️⃣7️⃣ More Rationalising Examples

Example 28: Rationalise 3/√5

Example 29: Rationalise 5/√3

Example 30: Rationalise 6/√2

🌍 Real-Life Example: Diagonal of a Square

Example 31

🌍 Real-Life Example: Square Garden

Example 32

🎯 Exam Tips

⚠ Common Mistakes

🎮 Your Turn!

🟢 Foundation Questions

🔵 Developing Questions