1. Manipulation of Surds

2. For any positive numbers a and b, can you observe the relationship between
Can you evaluate the following expressions?
 36 6
2 3 6
100 10
2 5 10
144 12
3 4 12
225 15
3 5 15

3. Properties of Surds (I)
This property is useful in evaluating square roots.
For example,

4. Follow-up question
Evaluate the following expressions without using a calculator.
Solution

5. For any positive numbers a and b, can you observe the relationship between
Now, try to evaluate the following expressions. 4 2 2
16 4 4

6. Properties of Surds (II)
We can also use this property to evaluate square roots.
For example,

7. Follow-up question
Evaluate the following expressions without using a calculator.
Solution

9. Rationalization of Denominators
For example, .
the
called is 2)
(e.g.
number
rational a
into ) 2
irrational an
from
denominator
changing of
process
The r
denominato
ation
rationaliz

10. Follow-up question
Rationalize the denominators of the following expressions.
Solution

11. Surd in its Simplest Form
For a surd with an integer inside the radical sign, we can simplify the surd by using its properties. For example,
The surd obtained is said to be in its simplest form.
i.e. 25 is the simplest form of 20.

12. To simplify a surd with a fraction inside the radical sign, we have to rationalize denominator first. For example,

13. For surds in their simplest form,15
For surds in their simplest form,
the number inside each radical sign does not have a square number (other than 1) as its factor.

14. Follow-up question
Determine whether each of the following surds is in its simplest form. If not, express it in its simplest form.
Solution

15. Like surds and unlike surds
Some surds contain the same number inside the radical signs when expressed in their simplest forms. They are called like surds.
For example, 3
,3 and contain
3.
 They are like surds.

16. Are they like surds?
Let’s simplify them first.
 They are like surds.

17. However, unlike surds cannot be combined.
called
are
They
signs.
radical
the
inside
number
same
contain
not do 7 4
and 5 2
surds
unlike
However, unlike surds cannot be combined.

18. Addition, Subtraction, Multiplication and Division of Surds
In adding or subtracting surds, we should express all the surds in their simplest forms and then combine the like surds.
For example,

19. In multiplication and division of surds, we can applied the properties of surds to do the calculation.
For example,

20. Follow-up question
Simplify the following expressions.
Solution