Surds

Chapter

4

Radicals (Surds)

cyan

magenta

yellow

95

50

75

100

Radicals on a number line Operations with radicals Expansions with radicals Division by radicals

A B C D

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

Contents:

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\087IB_MYP4_04.CDR Friday, 29 February 2008 4:01:39 PM PETERDELL

IB MYP_4

88

RADICALS (SURDS) (Chapter 4)

INTRODUCTION In previous years we used the Theorem of Pythagoras to find the length of the third side of a triangle. p p Our answers often involved radicals such as 2, 3, p 5, and so on.

~`5

1

2

A radical is a number that is written using the radical sign p . p p p p Radicals such as 4 and 9 are rational since 4 = 2 and 9 = 3. p p p Radicals such as 2, 3 and 5 are irrational. They have decimal expansions which neither terminate nor recur. Irrational radicals are also known as surds.

RESEARCH ² Where did the names radical and surd come from? ² Why do we use the word irrational to describe some numbers? ² Before we had calculators and computers, finding decimal representations for numbers like p12 to four or five decimal places was quite difficult and time consuming. 1 Imagine having to find 1:414 correct to five 21 decimal places using long division! A method was devised to do this calculation quickly. What was the process?

SQUARE ROOTS p p p The square root of a or a is the positive number which obeys the rule a £ a = a. p For a to have meaning we require a to be non-negative, i.e., a > 0. p p p 5 £ 5 = 5 or ( 5)2 = 5. For example, p Note that 4 = 2, not §2, since the square root of a number cannot be negative.

A

RADICALS ON A NUMBER LINE

p If we convert a radical such as 5 to a decimal we can find its approximate position on a p p 5 ¼ 2:236 067, so 5 is close to 2 14 . number line. ~`5

cyan

magenta

yellow

3

95

100

50

75

25

0

5

95

100

50

2

75

25

0

5

95

100

50

75

1

25

0

5

95

100

50

75

25

0

5

0

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\088IB_MYP4_04.CDR Wednesday, 5 March 2008 9:14:46 AM PETERDELL

IB MYP_4

89

RADICALS (SURDS) (Chapter 4)

p We can also construct the position of 5 on a number line using a ruler and compass. Since p p 12 + 22 = ( 5)2 , we can use a right-angled triangle with sides of length 1, 2 and 5. Step 1:

Draw a number line and mark the numbers 0, 1, 2, and 3 on it, 1 cm apart.

Step 2:

With compass point on 1, draw an arc above 2. Do the same with compass point on 3 using the same radius. Draw the perpendicular at 2 through the intersection of these arcs, and mark off 1 cm. Call this point A.

1 ~`5 0

1

2

p 5 cm.

Step 3:

Complete the right angled triangle. Its sides are 2, 1 and

Step 4:

With centre O and radius OA, draw an arc through A to meet the p number line. It meets the number line at 5.

3

DEMO

EXERCISE 4A

p 1 Notice that 12 + 42 = 17 = ( 17)2 . p Locate 17 on a number line using an accurate construction. a The sum of the squares of which two positive integers is 13? p b Accurately construct the position of 13 on a number line. p 3 Can we construct the exact position of 6 on a number line using the method above? 2

4 7 cannot be written as the sum of two squares so the above method cannot be used for locating 4 p ~`7 7 on the number line. p 3 However, 42 ¡32 = 7, so 42 = 32 +( 7)2 . p We can thus construct a right angled triangle with sides of length 4, 3 and 7. p Use such a triangle to accurately locate 7 on a number line.

B

OPERATIONS WITH RADICALS

ADDING AND SUBTRACTING RADICALS We can add and subtract ‘like radicals’ in the same way as we do ‘like terms’ in algebra. p p p For example: ² just as 3a + 2a = 5a, 3 2 + 2 2 = 5 2 p p p ² just as 6b ¡ 4b = 2b, 6 3 ¡ 4 3 = 2 3. Example 1

cyan

magenta

yellow

p p p p 7 ¡ 2(1 ¡ 7) = 7 ¡ 2 + 2 7 p =3 7¡2

95

100

50

75

25

0

5

95

100

50

75

25

b

0

p p p a 3 2 ¡ 4 2 = ¡1 2 p =¡ 2

5

95

p p b 7 ¡ 2(1 ¡ 7)

100

50

75

25

0

5

95

100

50

75

25

0

5

Simplify:

Self Tutor p p a 3 2¡4 2

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\089IB_MYP4_04.CDR Wednesday, 5 March 2008 9:19:04 AM PETERDELL

IB MYP_4

90

RADICALS (SURDS) (Chapter 4)

SIMPLIFYING PRODUCTS p p p a a = ( a)2 = a p p p a b = ab r p a a p = b b

We have established in previous years that:

Example 2

p a ( 2)2

Simplify:

a

Self Tutor µ

p b ( 2)3

p ( 2)2 p p = 2£ 2 =2

c

¶2

µ

p ( 2)3 p p p = 2£ 2£ 2 p =2 2

b

4 p 2 c

4 p 2

¶2

42 = p ( 2)2 =

16 2

=8

Example 3

Self Tutor p a (3 2)2

Simplifying:

p p b 3 3 £ (¡2 3)

p (3 2)2 p p =3 2£3 2 =9£2 = 18

a

Example 4

Self Tutor

Write in simplest form: p p a 2£ 5

cyan

p p b 3 2 £ 4 11

magenta

With practice you should not need the middle steps.

yellow

95

100

50

75

25

0

5

95

p p 3 2 £ 4 11 p p = 3 £ 4 £ 2 £ 11 p = 12 £ 2 £ 11 p = 12 22

100

50

75

25

0

5

95

b

100

50

75

25

0

p p 2£ 5 p = 2£5 p = 10

5

95

100

50

75

25

0

5

a

p p 3 3 £ (¡2 3) p p = 3 £ ¡2 £ 3 £ 3 = ¡6 £ 3 = ¡18

b

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\090IB_MYP4_04.CDR Wednesday, 5 March 2008 9:24:17 AM PETERDELL

IB MYP_4

RADICALS (SURDS) (Chapter 4)

Example 5

Self Tutor a

Simplify:

a = =

91

p 75 p 3

p 32 p 2 2

b

p 75 p 3 q

b

75 3

=

p 25

= =

=5

p 32 p 2 2 q 1 2 1 2 1 2

32 2

p 16 £4

=2

EXERCISE 4B.1 1 Simplify: p p a 3 2+7 2 p p c 6 5¡7 5 p p e 3 ¡ (2 ¡ 3) p p p p g 5 2¡ 3+ 2¡ 3 p p p i 3 3 ¡ 2 ¡ (1 ¡ 2) p p p p k 3( 3 ¡ 2) ¡ ( 2 ¡ 3)

p p b 11 3 ¡ 8 3 p p d ¡ 2+2 2 p p f ¡ 2 ¡ (3 + 2) p p p p h 7¡2 2+ 7¡ 2 p p j 2( 3 + 1) + 3(1 ¡ 3) p p l 3( 3 ¡ 1) ¡ 2(2 ¡ 3)

2 Simplify: p a ( 3)2

p b ( 3)3

p c ( 3)5

p e ( 7)2

p f ( 7)3

p i ( 5)2

p j ( 5)4

3 Simplify: p a (2 2)2 p d (3 3)2 p g (2 7)2 p p j 3 2£4 2 p m (¡4 2)2 p p p (¡2 3)(¡5 3)

µ k

yellow

5 p 5

black

95

Y:\HAESE\IB_MYP4\IB_MYP4_04\091IB_MYP4_04.CDR Thursday, 13 March 2008 11:42:19 AM PETERDELL

d

¶2

µ h

¶2

q 2 14 25

100

50

75

25

0

c

5

95

q 1 79

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

b

magenta

g

1 p 7

p b (4 2)2 p e (2 5)2 p h (2 10)2 p p k 5 3£2 3 p n (¡7 3)2 p p q (¡2 7) £ 3 7

4 Simplify: q a 6 14

cyan

µ

µ

µ l

1 p 3 3 p 7 10 p 5

¶2 ¶2 ¶2

p c (2 3)2 p f (3 5)2 p i (7 10)2 p p l 7 2£5 2 p p o 2 £ (¡3 2) p p r 11 £ (¡2 11)

d

q 7 19

IB MYP_4

92

RADICALS (SURDS) (Chapter 4)

5 Simplify: p p a 2£ 3 p p d 7£ 3 p p g 5 2£ 7 p p j (¡ 7) £ (¡2 3) 6 Simplify: p 8 a p 2 p 75 e p 5 p 3 6 i p 2

p p 2£ 7 p p e 2 2£5 3 p p h 2 6£3 5 p p k (2 3)2 £ 2 5

p p 2 £ 17 p f (3 2)2 p p i ¡5 2 £ 2 7 p p l (2 2)3 £ 5 3

b

p 3 b p 27 p 5 f p 75 p 4 12 j p 3

c

p 18 c p 3 p 18 g p 2 p 4 6 k p 24

p 2 d p 50 p 3 h p 60 p 3 98 p l 2 2

p p p p p p 9 + 16 = 9 + 16 ? Is 25 ¡ 16 = 25 ¡ 16 ? p p p p p p b Are a + b = a + b and a ¡ b = a ¡ b possible laws for radical numbers?

7

a Is

p p p a Prove that a b = ab for all positive numbers a and b. p p p Hint: Consider ( a b)2 and ( ab)2 . r p a a for a > 0 and b > 0. b Prove that p = b b

8

SIMPLEST RADICAL FORM A radical is in simplest form when the number under the radical sign is the smallest possible integer.

Example 6 p Write 8 in simplest form.

We look for the largest perfect square that can be taken out as a factor of this number.

Self Tutor p 8 p = 4£2 p p = 4£ 2 p =2 2

cyan

magenta

yellow

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

p p p p p 32 = 4 £ 8 = 2 8 is not in simplest form as 8 can be further simplified into 2 2. p p In simplest form, 32 = 4 2.

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\092IB_MYP4_04.CDR Wednesday, 5 March 2008 9:25:31 AM PETERDELL

IB MYP_4

RADICALS (SURDS) (Chapter 4)

Example 7 p Write 432 in simplest radical form.

Self Tutor

93

It may be useful to do a prime factorisation of the number under the radical sign.

p 432 p = 24 £ 33 p p = 24 £ 33 p =4£3 3 p = 12 3

EXERCISE 4B.2

p 1 Write in the form k 2 where k is an integer: p p a 18 b 50 c p p e 162 f 200 g p 2 Write in the form k 3 where k is an integer: p p a 12 b 27 c p 3 Write in the form k 5 where k 2 Z : p p a 20 b 80 c 4 Write in simplest radical form: p p a 99 b 52 p p e 48 f 125 p p i 176 j 150

p 72 p 20 000

d

p 48

d

p 300

p 320

d

p 500

p 40 p g 147 p k 275 c

p 98 p h 2 000 000

p 63 p h 175 p l 2000

d

p 5 Write in simplest radical form a + b n where a, b 2 Q , n 2 Z : p p p p 4+ 8 6 ¡ 12 4 + 18 8 ¡ 32 a b c d 2 2 4 4 p p p p 12 + 72 18 + 27 14 ¡ 50 5 ¡ 200 e f g h 6 6 8 10

C

EXPANSIONS WITH RADICALS

The rules for expanding radical expressions containing brackets are identical to those for ordinary algebra. a(b + c) = ab + ac (a + b)(c + d) = ac + ad + bc + bd (a + b)2 = a2 + 2ab + b2

cyan

magenta

yellow

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

(a + b)(a ¡ b) = a2 ¡ b2

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\093IB_MYP4_04.CDR Wednesday, 5 March 2008 9:26:42 AM PETERDELL

IB MYP_4

94

RADICALS (SURDS) (Chapter 4)

Example 8

a 2(2 +

Simplify: a

Self Tutor p 3)

b

p 2(2 + 3) p =2£2+2£ 3 p =4+2 3

p p 2(5 ¡ 2 2)

b

Example 9

Self Tutor

Expand and simplify: p p a ¡ 3(2 + 3) a

With practice you should not need the middle steps.

p p 2(5 ¡ 2 2) p p p = 2 £ 5 + 2 £ ¡2 2 p =5 2¡4

p p p b ¡ 2( 2 ¡ 3)

p p ¡ 3(2 + 3) p p p =¡ 3£2+¡ 3£ 3 p = ¡2 3 ¡ 3

p p p ¡ 2( 2 ¡ 3) p p p p =¡ 2£ 2+¡ 2£¡ 3 p = ¡2 + 6

b

EXERCISE 4C 1 Expand and simplify: p a 4(3 + 2) p d 6( 11 ¡ 4) p p g 3(2 + 2 3) p p j 5(2 5 ¡ 1) 2 Expand and simplify: p p a ¡ 2(4 + 2) p p d ¡ 3(3 + 3) p p p g ¡ 5(2 2 ¡ 3) p p j ¡ 7(2 7 + 4)

p p b 3( 2 + 3) p p e 2(1 + 2) p p p h 3( 3 ¡ 2) p p p k 5(2 5 + 3)

p c 5(4 ¡ 7) p p f 2( 2 ¡ 5) p p i 5(6 ¡ 5) p p p l 7(2 + 7 + 2)

p p 2(3 ¡ 2) p p e ¡ 3(5 ¡ 3) p p p h ¡2 2( 2 + 3) p p k ¡ 11(2 ¡ 11)

p p p c ¡ 2( 2 ¡ 7) p p p f ¡ 3(2 3 + 5) p p i ¡2 3(1 ¡ 2 2) p p l ¡( 2)3 (4 ¡ 2 2)

b

Example 10

Self Tutor

cyan

magenta

yellow

p p 5)(1 ¡ 5)

95

50

p p (3 + 5)(1 ¡ 5) p p = (3 + 5)(1 + ¡ 5) p p p = (3 + 5)1 + (3 + 5)(¡ 5) p p =3+ 5¡3 5¡5 p = ¡2 ¡ 2 5

75

25

0

5

95

b

100

50

75

25

0

5

95

100

50

75

25

0

p p (2 + 2)(3 + 2) p p p = (2 + 2)3 + (2 + 2) 2 p p =6+3 2+2 2+2 p =8+5 2

5

95

100

50

75

25

0

5

a

b (3 +

100

Expand and simplify: p p a (2 + 2)(3 + 2)

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\094IB_MYP4_04.CDR Wednesday, 5 March 2008 10:23:22 AM PETERDELL

IB MYP_4

RADICALS (SURDS) (Chapter 4)

3 Expand and simplify: p p a (2 + 2)(3 + 2) p p c ( 2 + 2)( 2 ¡ 1) p p e (2 + 3)(2 ¡ 3) p p g ( 7 + 2)( 7 ¡ 3) p p i (3 2 + 1)(3 2 + 3)

p p 2)(3 + 2) p p (4 ¡ 3)(3 + 3) p p (2 ¡ 6)(5 + 6) p p p p ( 11 + 2)( 11 ¡ 2) p p (6 ¡ 2 2)(2 + 2)

b (3 + d f h j

Example 11

Self Tutor

Expand and simplify: p a ( 2 + 3)2 a

p p b ( 5 ¡ 3)2

p ( 2 + 3)2 p p = ( 2)2 + 2 2(3) + 32 p =2+6 2+9 p = 11 + 6 2

4 Expand and simplify: p a (1 + 3)2 p d (1 + 7)2 p p g ( 3 + 5)2 p j (2 2 + 3)2

p p ( 5 ¡ 3)2 p p = ( 5 + ¡ 3)2 p p p p = ( 5)2 + 2 5(¡ 3) + (¡ 3)2 p = 5 ¡ 2 15 + 3 p = 8 ¡ 2 15

b

p b ( 2 + 5)2 p p e ( 3 ¡ 2)2 p h (3 ¡ 6)2 p k (3 ¡ 2 2)2

p 2 2) p 2 f (4 ¡ 5) p p i ( 6 ¡ 3)2 p l (3 ¡ 5 2)2

c (3 ¡

Example 12

Self Tutor

Expand and simplify: p p a (4 + 2)(4 ¡ 2)

p p b (2 2 + 3)(2 2 ¡ 3)

p p (4 + 2)(4 ¡ 2) p = 42 ¡ ( 2)2 = 16 ¡ 2 = 14

magenta

yellow

95

50

75

25

0

p p b ( 3 ¡ 1)( 3 + 1) p p d ( 3 ¡ 4)( 3 + 4) p p f (2 + 5 2)(2 ¡ 5 2) p p h (2 5 + 6)(2 5 ¡ 6)

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

5 Expand and simplify: p p a (3 + 2)(3 ¡ 2) p p c (5 + 3)(5 ¡ 3) p p e ( 7 ¡ 3)( 7 + 3) p p p p g ( 7 ¡ 11)( 7 + 11)

cyan

p p (2 2 + 3)(2 2 ¡ 3) p = (2 2)2 ¡ 32 =8¡9 = ¡1

b

100

a

95

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\095IB_MYP4_04.CDR Friday, 29 February 2008 4:51:00 PM PETERDELL

IB MYP_4

96

RADICALS (SURDS) (Chapter 4)

p p i (3 2 + 2)(3 2 ¡ 2) p p p p k ( 3 ¡ 7)( 3 + 7)

p p p p j ( 3 ¡ 2)( 3 + 2) p p l (2 2 + 1)(2 2 ¡ 1)

D

DIVISION BY RADICALS 6 p 2

In numbers like

9 p p 5¡ 2

and

we have divided by a radical.

It is customary to ‘simplify’ these numbers by rewriting them without the radical in the denominator.

INVESTIGATION 1

DIVISION BY

p a

b p where a and a b are real numbers. To remove the radical from the denominator, there are two methods we could use:

In this investigation we consider fractions of the form

² ‘splitting’ the numerator

² rationalising the denominator

What to do: 6 p . 2

1 Consider the fraction

a Since 2 is a factor of 6, ‘split’ the 6 into 3 £ 6 b Simplify p . 2

p p 2 £ 2.

7 2 Can the method of ‘splitting’ the numerator be used to simplify p ? 2 7 p . 2

3 Consider the fraction

p 2 p , are we changing its value? 2

a If we multiply this fraction by b Simplify

7 p 2

by multiplying both its numerator and denominator by

p 2.

4 The method in 3 is called ‘rationalising the denominator’. Will this method work b where a and b are real? for all fractions of the form p a

cyan

magenta

yellow

Y:\HAESE\IB_MYP4\IB_MYP4_04\096IB_MYP4_04.CDR Monday, 3 March 2008 9:19:26 AM PETERDELL

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

b From the Investigation above, you should have found that for any fraction of the form p , a p p a a we can remove the radical from the denominator by multiplying by p . Since p = 1, a a we do not change the value of the fraction.

black

IB MYP_4

RADICALS (SURDS) (Chapter 4)

Example 13

Self Tutor

Multiplying the original p 3 or number by p 3 p 7 does not change p 7 its value.

Write with an integer denominator: 6 35 a p b p 3 7 a

6 p 3 p 3 6 =p £p 3 3 p 2 6 3 = 31 p =2 3

97

35 p 7 p 7 35 =p £p 7 7 p 5 35 7 = 7 1 p =5 7

b

EXERCISE 4D.1 1 Write with integer denominator: 1 a p 3

3 b p 3

9 c p 3

2 f p 2 5 k p 5 7 p p 7

6 g p 2 15 l p 5 21 q p 7

12 h p 2 ¡3 m p 5 2 r p 11

d i n s

11 p 3 p 3 p 2 200 p 5 26 p 13

p 2 e p 3 3 1 p 4 2 1 o p 3 5 1 t p ( 3)3 j

RADICAL CONJUGATES

p p Radical expressions such as 3 + 2 and 3 ¡ 2 which are identical except for opposing signs in the middle, are called radical conjugates. The radical conjugate of a +

p p b is a ¡ b.

INVESTIGATION 2

RADICAL CONJUGATES

c p can also be simplified to remove the a+ b radical from the denominator. To do this we use radical conjugates. Fractions of the form

What to do: 1 Expand and simplify: p p a (2 + 3)(2 ¡ 3)

p p b ( 3 ¡ 1)( 3 + 1)

cyan

magenta

95

yellow

Y:\HAESE\IB_MYP4\IB_MYP4_04\097IB_MYP4_04.CDR Friday, 11 April 2008 4:41:01 PM PETERDELL

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

2 What do you notice about your results in 1?

black

IB MYP_4

98

RADICALS (SURDS) (Chapter 4)

3 Show that for any integers a and b, the following products are integers: p p p p a (a + b)(a ¡ b) b ( a ¡ b)( a + b) 4

a Copy and complete: To remove the radicals from the denominator of a fraction, we can multiply the denominator by its ...... b What must we do to the numerator of the fraction to ensure we do not change its value?

From the Investigation above, we should have found that: to remove the radicals from the denominator of a fraction, we multiply both the numerator and the denominator by the radical conjugate of the denominator.

Example 14

Write

Self Tutor

14 p 3¡ 2

with an integer denominator. ¶Ã

p ! 3+ 2 p 3+ 2 p 14 = £ (3 + 2) 9¡2 p = 2(3 + 2) p =6+2 2

14 p = 3¡ 2

µ

14 p 3¡ 2

Example 15

cyan

a with integer denominator p b in the form a + b 2 where a, b 2 Q .

yellow

95

100

50

0

5

95

100

50

75

25

0

5

95

50

75

25

0

100

magenta

75

p 5¡ 2 b = 23 So, a =

1 p 5+ 2 p ! µ ¶ Ã 1 5¡ 2 p p £ = 5+ 2 5¡ 2 p 5¡ 2 = 25 ¡ 2 p 5¡ 2 = 23

5

95

100

50

75

25

0

5

a

1 p : 5+ 2

25

Write

Self Tutor

black

Y:\HAESE\IB_MYP4\IB_MYP4_04\098IB_MYP4_04.CDR Thursday, 13 March 2008 2:11:59 PM PETERDELL

5 23 5 23

¡

1 23

p 2

1 and b = ¡ 23 .

IB MYP_4

RADICALS (SURDS) (Chapter 4)

99

EXERCISE 4D.2 1 Write with integer denominator: 1 p a 3+ 2 p 1+ 2 p e 1¡ 2

2 1 p p b c 3¡ 2 2+ 5 p p 3 ¡2 2 p p f g 4¡ 3 1¡ 2 p 2 Write in the form a + b 2 where a, b 2 Q : p 3 4 2 p a p b c p 2¡3 2+ 2 2¡5 p 3 Write in the form a + b 3 where a, b 2 Q : p 4 6 3 p p a b p c 1¡ 3 3+2 2¡ 3 p p 4 a If a, b and c are integers, show that (a + b c)(a ¡ b c) is

p 2 p d 2¡ 2 p 1+ 5 p h 2¡ 5 p ¡2 2 d p 2+1 p 1+2 3 p d 3+ 3

an integer. p 2 p . 3 2¡5

1 p ii 1+2 3 p p p p a If a and b are integers, show that ( a + b)( a ¡ b) is also an integer.

b Write with an integer denominator: 5

i

b Write with an integer denominator:

p 3 p ii p 3¡ 5

1 p i p 2+ 3

HOW A CALCULATOR CALCULATES RATIONAL NUMBERS

LINKS click here

Areas of interaction: Human ingenuity

REVIEW SET 4A 1 Simplify: p a (2 3)2

µ b

4 p 2

¶2

p p c 3 2£2 5

a Copy and complete: 12 + 32 = (::::::)2

2

b Use a to accurately construct the position of and compass.

cyan

magenta

yellow

95

50

75

25

0

100 black

Y:\HAESE\IB_MYP4\IB_MYP4_04\099IB_MYP4_04.CDR Thursday, 13 March 2008 11:45:11 AM PETERDELL

q 12 14

p 10 on a number line using a ruler

p 35 c p 5

5

95

100

50

75

25

0

p 35 b p 7

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

3 Simplify: p 15 a p 3

d

p 2 d p 20

IB MYP_4

100

RADICALS (SURDS) (Chapter 4)

p 8 in simplest radical form. p p b Hence, simplify 5 2 ¡ 8 . p 5 Write 98 in simplest radical form.

4

a Write

6 Expand and simplify: p a 2( 3 + 1) p d (2 ¡ 5)2

p p 2(3 ¡ 2) p p e (3 + 2)(3 ¡ 2)

p 2 7) p p f (3 + 2)(1 ¡ 2)

c (1 +

b

7 Write with an integer denominator: p 3+2 10 b p a p 5 3+1 p 8 Write in the form a + b 5 where a, b 2 Q : p 2 5 3 p b p a 2¡ 5 5+1

p 1+ 7 p c 1¡ 7

REVIEW SET 4B 1 Simplify:

p q p 8 b p d 5 49 c (3 5)2 2 p Find the exact position of 12 on a number line using a ruler and compass construction. Explain your method. Hint: Look for two positive integers a and b such that a2 ¡ b2 = 12. p p 21 3 Simplify: a p b p 3 24 p p 3 ¡ 27 Simplify: p p 12 b 63 Write in simplest radical form: a p p 3 2 a

2

3 4 5

6 Expand and simplify: p a 3(2 ¡ 3) p p d ( 3 + 2)2

p p 7( 2 ¡ 1) p p e (2 ¡ 5)(2 + 5)

p 2 2) p p f (2 + 3)(3 ¡ 3)

c (3 ¡

b

7 Write with integer denominator:

p 1+ 2 p b 2¡ 2

24 a p 3

p 4¡ 5 p c 3+ 5

cyan

magenta

yellow

Y:\HAESE\IB_MYP4\IB_MYP4_04\100IB_MYP4_04.CDR Monday, 3 March 2008 9:31:13 AM PETERDELL

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

95

100

50

75

25

0

5

p 8 Write in the form a + b 3 where a, b 2 Q : p ¡ 3 18 p p b a 5¡ 3 3+ 3

black

IB MYP_4