3. Surds The square roots of most numbers cannot be found exactly.
For example, the value of √3 cannot be written exactly as a fraction or a decimal.
The value of √3 is an irrational number.
For this reason, it may be better – and more mathematically accurate – to leave the square root sign in.
Teacher notes
9 is not a surd because it can be written exactly: 3. All of the other examples given do not have an exact value. They are all irrational numbers. You could extend this to get students thinking of other irrational numbers that they have encountered. The most obvious example being the value of pi which students should be familiar with from their knowledge of Geometry.
Another example of a surd that they may be less familiar with, but that they will be able to relate to and can even test is the ratio of the length to the width of A4 paper which is in the ratio √2 : 1.
Photo credit: © Barry Barnes, Shutterstock.com
√3 is an example of a surd.
Which one of the following is not a surd: √2, √6 , √9 or √14? Why?

4. Multiplying surds
Surds can sometimes be difficult to work with, so it is important that we have guidelines on how to deal with them.
How would you calculate the value of √3 × √3?
We can think of this as squaring the square root of three.
Squaring and square rooting are inverse operations so: √3 × √3 = 3.
Teacher notes
Using the above result, √3 × √3 × √3 = 3 × √3 = 3√3. At this point, you may wish to emphasise to students that, like in algebra, we do not use the × sign when writing surds.
As a general rule, we can state that:
√a × √a = a
What is the value of √3 × √3 × √3?

5. Multiplying surds Using your calculator, find the value of √2 × √8.
What do you notice about the answer?
√2 × √8 = 4
(= √16)
4 is the square root of 16 and 2 × 8 = 16.
As a general rule, we can state that:
√a × √b = √ab
Teacher notes
Pupils should notice that although √2 and √8 are irrational numbers, their product is a rational number.
Similarly the product of √3 and √12 is a rational number.
√3 × √12 = 6 = √36.
6 is the square root of 36 and 3 × 12 = 36.
This is because 2 × 8 is a square number, as is 3 × 12. Use these examples to establish the general result.
√20 × √5 would be another example of two surds that follow this pattern. Ask pupils to give other examples of two surds that will multiply together to make a whole number.
What do you think the value of √3 × √12 will be? Why?
Use a calculator to check your result. Were you correct?
Can you find two other surds that follow this pattern?

6. √ Dividing surds Using your calculator, find the value of √20 ÷ √5.
What do you notice about the answer?
√20 ÷ √5 = 2
(= √4)
2 is the square root of 4 and 20 ÷ 5 = 4. √ a b
As a general rule, we can state that:
√a ÷ √b =
Teacher notes
Pupils should notice that although √20 and √5 are irrational numbers, their quotient is a rational number.
Similarly the quotient of √18 and √2 is a rational number. This is because 20 ÷ 5 is a square number, as is 18 ÷ 2 .
√18 ÷ √2 = 3 = √9.
3 is the square root of 9 and 18 ÷ 2 = 9.
√160 ÷ √10 would be another example of two surds that follow this pattern. There is a chance to practice multiplying and dividing surds on the two following interactive slides. Ask pupils to give you examples of other surds that divide to give whole numbers.
What do you think the value of √18 ÷ √2 will be? Why?
Use a calculator to check your result. Were you correct?
Can you find two other surds that follow this pattern?

7. Surd laws

8. Surds and rational numbers

10. Simplifying surds
When solving mathematical problems, we are often required to simplify surds by writing them in the form: a√b.
Can you simplify √50 by writing it in the form a√b?
Start by finding the largest square number that divides into 50.
We can use this to write:
√50 = √(25 × 2)
Teacher notes
Stress that when a surd is written in its simplest form, the number under the square root sign must not contain any factors that are square numbers.
= √25 × √2
= 5√2

11. Simplifying surds Teacher notes
For each example, ask pupils to give you the highest square number that divides into each surd before simplifying it.

12. Simplifying surds

14. Adding and subtracting surds
Surds can be added or subtracted if the number under the square root sign is the same.
Can you simplify the calculation: √27 + √75?
Start by writing √27 and √75 in their simplest forms.
√27 = √(9 × 3)
√75 = √(25 × 3)
= √9 × √3
= √25 × √3
Teacher notes
Compare this to adding like terms in algebra.
= 3√3
= 5√3
√27 + √75 = 3√3 + 5√3 =
8√3

15. Rationalizing the denominator
When a fraction has a surd as a denominator, we usually rewrite it so that the denominator is a rational number.
This process is called rationalizing the denominator. 5
Can you simplify the fraction ? √2
If we multiply the numerator and the denominator by the same number, the value of the fraction remains unchanged.
×√2
Multiply the numerator and the denominator by √2 to make the denominator into a whole number. 5 √2
5√2 = 2
×√2

16. Rationalizing practice
Teacher notes
Ask pupils to suggest what the numerator and denominator of each fraction should be multiplied by before revealing the solutions.

17. A rational result Teacher notes
You may wish to encourage students to try and explain why this happens using algebra: (a + √b)(a – √b) = a2 + a√b – a√b – b = a2 – b. The surd part has been removed (rationalized).

18. More complex work
Rationalizing more complex denominators requires a little bit of extra work in order to get a simplified answer. 5
6 + √11
How can we simplify the fraction ?
6 + √11 5
6 – √11
By multiplying both sides by 6 – √11, we will be able to remove the surd and thus rationalize the denominator. ×
30 – 5√11
36 – 11 =
30 – 5√11 25
Tidy the fraction to express it in its simplest form. =

19. Ab-surd-ly difficult? Teacher notes
Encourage students to: multiply the top and bottom by the same thing, simplify the top and bottom and then tidy the fraction at the end to express it in its simplest form.

21. Perimeter and area
The following rectangle has been drawn on a centimetre grid.
Teacher notes
Talk through the application of Pythagoras’ Theorem to find the width and length of the rectangle.
Width (x) = √( ) = √(9 + 1) = √10 units
Length (y) = √( ) = √(36 + 4) = √40 = 2√10 units. Encourage students to use the rules of surds to simplify √40 to 2√10.
Perimeter = √10 + 2√10 + √10 + 2√10 = 6√10 units
Area = √10 × 2√10 = 2 × √10 × √10 = 2 × 10 = 20 units2
Verify that the area of the rectangle is 20 units squared by completing the surrounding 5 by 7 rectangle and subtracting the area of the four surrounding triangles.
Point out that had we evaluated the surds we could not have been sure we had an exact answer, because we would have had to have rounded at that stage.
Use the given lengths to find the length (y) and width (x) of the rectangle. Show your working.
Find its perimeter and area in surd form.

22. The Pythagorean spiral
An architect is working on the design for a spiral staircase.
He thinks he can use Pythagoras’ theorem to calculate the missing lengths.
Can you calculate the width of the sections marked: a, b, c, d, e and f? Express your answers in surd form wherever necessary.
Draw the diagram to measure the lengths and see how accurate you are.
Teacher notes
The lengths are the consecutive hypotenuses of each subsequent right angled triangle. The lengths on the outside of the spiral are all 1. The calculated lengths are: √2, √3, √4 = 2, √5, √6 and √7. Students may need to be reminded of Pythagoras Theorem before starting this.
a² = 1² + 1² = = 2
a = √2
b² = √2² + 1² = = 3
b = √3
c² = √3² + 1² = = 4
c = √4 = 2
d² = 2² + 1² = = 5
d = √5
e² = √5² + 1² = = 6
e = √6
f² = √6² + 1² = = 7
f = √7

23. Laying paving slabs
A landscape gardener has drawn up plans for laying paving slabs through a garden.
The plans have not been drawn to scale.
The slabs need to be accurately cut to fit within the pattern perfectly.
Without using a calculator, work out the lengths: a, b, c, d, and e to help you find x.
Make sure all your answers are in surd form and show your working.
Teacher notes
Students should be encouraged to leave all answers as surds (so no calculators), and simplify where possible. Using Pythagoras’ theorem, we know that: a² = 2² + 3² = = 13
a = √13.
Once we have calculated a, we know that: b² = √13² + √11² = = 24
b = √24 = 2√6
Using the information from b, we know that: c² = b² + (2√5)² = (2√6)² + (2√5)² = √24² + √20² = = 44
c = √44 = 2√11
Knowing the value of c allows us to work out d: d² = c² – 6² = (2√11)² – 6² = √44² – 36 = 44 – 36 = 8
d = √8 = 2√2
Using our value of d, we can work out e: e² = (3√3)² – d² = (3√3)² – (2√2)² = √27² – √8² = 27 – 8 = 19
e = √19
Finally, we can find x: x² = e² – √11² = √19² – √11² = 19 – 11 = 8
x = √8 = 2√2