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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