SCHOOL OF SCIENCE AND TECHNOLOGY, SINGAPOREMATHEMATICS DEPARTMENT 2014SECONDARY 2 Fun Pack (Part 2)

Name: ( ) Class: S__–__

F.P.2 SURDS

1) IntroductionA number that cannot be expressed as a integer fraction is called an irrational number and the following are examples of irrational numbers:

2 , 33 , , e

An irrational number involving a root is called a surd and the following are examples of surds:

2) Rules involving SurdsThere are several rules involving surds:a)

b) ba

ba

c)

Surds adhere to the rules of algebra as well.

d)

e)

f)

g)

Note: Surds should be expressed in the simplest form. First, express the number in terms

of a square (largest possible) another whole number (which does not contain a perfect

square). Then you can apply rule (a) above to separate the two parts here.

Challenge: Can you use the Laws of Indices to prove the rules shown above?

Example 1 Simplify √20

√20=√22×5¿2√5

For All Questions, leave your answers exact and don’t round off.

Question 1 a) Express 48 as a product of its PRIME FACTORS,

b) Hence, simplify .

Question 2 Simplify the following expression.

Question 3 Simplify 7 2 .

Question 4 Simplify .

Question 5 Given that , find the value of a.

CONJUGATE SURDS

These are specially related surds whose product is rational:

are conjugate surds of each other.

What would happen if you were to MULTIPLY the 2 conjugate surds?

Notice the similarity between and .

Hence, conclude the IMPORTANCE of finding the CONJUGATE SURDS?

_____________________________________________________________

NOTE – convince yourself that the CONJUGATE of could be expressed

EITHER as OR as .

Question 6 Simplify .

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Question 7 Find the value of .

RATIONALISATION OF THE SURD

Looking at the following examples of rational and irrational surds, deduce the definition of a rational surd. You can compare the values by checking their values in the calculator.

Rational Surd Irrational Surd

A surd is considered as rational when _______________________________.

Example 2 Rationalise

√3√5+√2

Important: The key is to multiply the conjugate surd of the denominator on both the numerator and denominator of the fraction.

√3√5+√2

=√3 (√5−√2 )(√5+√2 ) (√5−√2 )

=√3×5−√3×2(√5 )2−(√2 )2

¿√15−√65−2

¿√15−√63

Question 8 Rationalise 133 .

Question 9 Simplify .

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Question 10 Given and . Identify the larger number without using a calculator.

Question 11 Given that , evaluate 2

2 1x

x .

Question 12 Simplify

1√1+√2

+ 1√2+√3

+ 1√3+√4

+. . .+ 1√n+√n+1 .

Question 13 Find the value of h and of k for each of the following:

a)

b)

(Hint: group (1−√2 ) first and rationalise the denominator twice)Question 14 The sides AB and BC of a triangle, right-angled at B, are of lengths

and respectively. Find,a) area of the triangleb) the value of AC2.

Answers:

Q1. a) 24×3 , b) 4 √3 . Q2. 14√2 Q3.

425 √3

Q4. 62−12√3 Q5. a=1 Q6. 3

Q7. 10 Q8.

3+√32 Q9. −√10−√3

Q10. Hint: use division to find if the rationalised fraction is greater than or less than 1.

Q11. 10 5

8+15

8 √21Q12. √n+1−1 Q13 a)

h=15

, k= 115 , b) h=1 ,k=1

Q14. a) 8+ 7

2 √10, b) 45+10√10

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