Surds ( )√ =√

These are irrational numbers which cannot be Multiplication of surds

expressed in terms of where a and b are Finding the product of two surd numbers is the
rational. Irrational numbers may be defined as same as finding the root of the product of two
square roots of prime numbers. numbers.
Examples are √ √ √ i.e. √ √ √
Expression of root of numbers in surd form Example 3
Example 1 Find the value of the following and give your
answers in the simplest form
Write the following as the simplest surds
(i) √ √
(i) √ ( )√ ( )√ ( )√
(ii) √ √
Solution (iii) ( √ √ )
(i) √ √ √ √ √ Solution
(ii) √ √ √ √ √
(iii) √ √ √ √ √ (i) √ √ √ √
(iv) √ √ √ √ √ (ii) √ √ √ √
(iii) ( √ √ )
Addition and subtraction of surds
Using Pascal’s triangle; the coefficients of the
This is done by expressing the surds in their
terms in the expansion (a + b)3 are 1 3 3 1
simplest form

Example 2 ( √ √ ) =

( √ ) ( √ ) ( √ ) ( √ )( √ ) ( √ )
(i) √ √ √
=( √ ) ( )( √ ) ( )( √ ) √
=√ √ √
= 162√ √
=√ √ √ √ √ √
Example 3
= √ √ √
Express in form b√ where b and
(5 – 9 + 4)√ = 0 √ √ √ √
c are integers. (05 marks)
(ii) √ √ √ √
(√ √ ) (√ √ )
=√ √ √ √ √ √ √ √ (√ √ )(√ √ )
= √ √ √ √ √

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 Hence b = 8 and c= 3 Example 5

Example 4 Rationalize the following

√ (i)
Show that √ √
(ii)
√
( )
√ √ (iii)
√

Solution
√
√ √ √
(i)
√ √ √
√ (ii)
√ √
√ √ √
√ √
(iii)
√ √ √ √

In the second case rationalize by multiplying the

√
= numerator and denominator by the conjugate of
the denominator.
Example 5
Note
√ √
Without using a calculator, evaluate .
√ √ (i) The conjugate a + √ is a - √ and that
a - √ is a + √
Solution
(ii) The product of a surd function and its
√ √ √ √ conjugate is equal to the difference of two
=
√ √ √ √
√ √ √ squares. i.e. (a + √ )( a - √ ) = (
=
√ √ √
√ √
(√ ) )
=
√
√ Example 7
=
√
=5 Rationalize the following

(i)
√

Division of surds (ii)

√
√
There are two types of division of surds; (iii)
√

- A fraction whose denominators has a single solution

term such as , etc. ( √ ) ( √ )
√ √
(i) ( √ )
- A fraction whose denominator has double √ ( √ )( √ )
√ ( √ ) ( √ ) ( √ )
terms, such as , , etc. (ii)
√ √ √ √ √ ( √ )(( √ ))
- In both cases, first eliminate the surds from √ √ ( ) (√ )
the denominator. The process of eliminating (iv)
√ √ (√ ) ( )
surds from the denominator is called
rationalization. Set square angle (300, 450 and 600)
- In the first case rationalize the fraction by (a) Consider an equilateral triangle ABC of each
multiplying the numerator and denominator side = a units
by the surd term of the denominator.

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 C √ √ √
(a) ( )( )
√ √
√

a 300 a √ √
√
√
600 600 (b) ( ) ( ) (
√
)
A B √ √
D a
√
√
√ √
( ) ( )( )
√ √
√ √ √
√
EC=

√ √
cos 600 = ; sin600 = Revision exercise
√ 1. Simplify
tan 600= √
(a) √ [4√ ]
√ √
cos 300 = ; sin600 = (b) √ [9√ ]
(c) √ √
tan 600= (d) √ √ ]
√ √
(e) √ [5√
(b) Given a right angled isosceles triangle ABC
(f) √ [ √ ]
with equal perpendicular sides each of
2. Simplify the following suds
length a units
(a) √ √ √ [2√
C (b) √ √ √ [18√
(c) √ √ √ √ √
450
a a√ [23√ √
3. Given that 4√ √ √ √ ,
450 find the value of x [-14]
B A
a 4. Find the value of the following simplifying
the answer as much as possible
√ (a) ( √ √ )( √ √ )[ √ ]
cos 45 = sin450 =
0
√
√ √ 5. Express each of the following in the form
0
tan 45 = where a, b and c are integers
√
Example 8 (a) * +
√
√
Express without a surd in the denominator each (b) * +
√
of the following √
(c) [ √ ]
√
(a)

(b) ( )

Solution

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Thanks

Dr. Bbosa Science

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