1.1 Prime factors, square roots and cube roots

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Tutorial 1: (Number Systems)

Total: 30

1. Simplify:

1.1 $\sqrt{\text{100}-\text{36}}$ [1]

1.2 $\sqrt{\frac{\text{25}}{\text{49}}}$ [1]

1.3 $\sqrt{{2}^{6}{3}^{\text{15}}}$ [2]

1.4 $\sqrt{9}\left(\sqrt{9}+\sqrt{\text{16}}\right)$ [3]

1.5 9² [1]

1.6 $\sqrt{a}=\text{4,}a=$ [1]

1.7 $\sqrt[3]{a}=\text{5,}a=$ [1] [10]

2. Use the 324, and answer the following questions:

2.1 Is 324 divisible by 3? Give a reason for your answer. [2]

2.2 Write 324 as the product of its prime factors [3]

 324

2.3 Now determine $\sqrt{\text{324}}$ [2]

2.4 Is 324 a perfect square? Give a reason for your answer. [2] [9]

3. Determine each of the following without using your calculator.

3.1 $\sqrt{\text{81}}$ [1]

3.2 $\sqrt{\frac{\text{36}}{4}}$ [2]

3.3 $\sqrt{{3}^{2}+{4}^{2}}$ [2]

3.4 $\sqrt{\text{16}{x}^{\text{16}}}$ [2]

4. If x = 3, determine:

4.1 ${4}^{x}$ [2]

4.2 $\sqrt[x]{\text{27}}$ [2] [11]

Tutorial

 I demonstrate knowledge and understanding of: Learning outcomes 0000 000 00 0 1. natural numbers ( N ) and whole numbers ( N 0 ) 1.1 2. the identification of the different types of numbers; 1.1 3. compound numbers; 1.2.6 4. divisibility rules; 1.2.6 5. the multiples of a number; 1.2.6 6. the factors of a number; 1.2.6 7. prime numbers; 1.1 8. prime factors; 1.2.6 9. expressing a number as the product of its prime factors; 1.2.6; 1.2.3 10. expressing prime factors in exponent notation; 1.2.3 11. even and odd numbers; 1.1 12. square roots of a number; 1.2.7 13. cube roots of a number; 1.2.7 14. the smallest common factor (LCM); 1.2.6 15. the biggest common divider (BCD). 1.2.6
 The learner’s … 1 2 3 4 work is… Not done.. Partially done. Mostly complete. Complete. layout of the work is… Not understandable. Difficult to follow. Sometimes easy to follow. Easy to follow. accuracy of calculations… Are mathematically incorrect. Contain major errors. Contain minor errors. Are correct.
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Test 1: (Number Systems)

Total: 30

1. Tabulate the following:

1.1 All the prime numbers between 20 and 30. [2]

1.2 All the factors of 12. [2]

1.3 All factors of 12 which are compound numbers [2] [6]

2. Determine the smallest natural number for * so that the following number is divisible by 3. (Give a reason for your answer)

1213156*3 [2]

3. Determine the following without using your calculator.

3.1 $\sqrt{\text{36}+\text{64}}$ [2]

3.2 $\sqrt[3]{{2}^{9}}$ [2]

3.3 $\sqrt{2\frac{7}{9}}$ [3]

3.4 $\sqrt{\text{0,04}}$ [2]

3.5 $\sqrt{\text{100}-\text{36}}$ [2]

3.6 $\sqrt[3]{8×\text{27}}$ [2]

3.7 $\left(\sqrt{9}{\right)}^{2}$ [2]

3.8 $\sqrt[3]{\text{64}-1}$ [2] [17]

4. Determine $\sqrt[3]{\text{1 728}}$ using prime factors, without using a calculator.

[5]

5. Bonus question

If (n) means n n what is the value of ((2)) ? [2]

Enrichment Exercise for the quick learner

(Learning unit 1)

Each question has five possible answers. Only one answer is correct. Place a cross (X) over the letter that indicates the correct answer.

1. If n and p are both odd, which of the following will be even?

a) np b) n ² p + 2 c) n + p +1 d) 2 n +3 p +5 e) 2 n + p

2. R 120 is divided amongst three men in the ratio 3 : 4: 9. The one with the smallest share will receive ...

a) R16 b) R20 c) R22,50 d) R24,50 e) R40

3. How many triangles are there in the figure?

a) 8 b) 12 c) 14 d) 16 e) 20

4. A decagon has 2 interior angles of 120° each. If all the remaining angles are of the same size, each angle will be equal to ...

a) 15° b) 30° c) 120° d) 150° e) 165°

5. The last digit of the number 3 1993 is ....

a) 1 b) 3 c) 6 d) 7 e) 9

6. The figure below has 5 squares. If AB = 6, the area of the figure is...

a) 12 b) 20 c) 24 d) 36 e) impossible

Assessment

 Learning outcomes(LOs) LO 1 Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. Assessment standards(ASs) We know this when the learner : 1.1 describes and illustrates the historical and cultural development of numbers; 1.2 recognises, classifies and represents the following numbers in order to describe and compare them:1.2.3 numbers written in exponent form; including squares and cubes of natural numbers and their square roots and cube roots;1.2.6 multiples and factors;1.2.7 irrational numbers in the context of measurement (e.g. square and cube roots on non-perfect squares and cubes); 1.6 estimates and calculates by selecting suitable steps for solving problems that involve the following:1.6.2 multiple steps with rational numbers (including division with fractions and decimals);1.6.3 exponents.

Class assignment 2

1.1 48 = 2 4 × 3; 60 = 2² × 3 × 5; 450 = 2 x 3² x 5²;

P 48 = {2, 3}; P 60 = {2, 3, 5}; P 450 = {2, 3, 5};

2.1 i)== (2 10 )

= 2 5

= 32

ii)== (2³ x 5³)

= 2 x 5

= 10

2.2 a) 36

b) 192

c) 1

d) 1

e) 2

f) 17

g) 63

h) 9

i) 10

j) 4

k) 27

l) 8 x 6

Homework assignment 2

1.1= (2 12 )

= 2 4

= 16

1.2= (2 4 x 3 4 )

= 2 x 3

= 6

2.1= 3² = 9

2.2 5a²b 5

2.3=x 3 =

= 1,2

2.4: 4 + 64 = 68

• :2(8) = 16
• :13

2.7 () 2 = 54

2.8= 36

• :2(9) = 18
• :9 - 27 = -18

Class assignment 3

21. LCM : Lowest common multiple

LCM of 2, 6, 12 :

24 HCF : Highest common factor

HCF of 24 and 48 :

2. 38 = 2 x 19

57 = 3 x 19

95 = 5 x 19

HCF = 19

LCM = 19 x 2 x 3 x 5

= 570

TUTORIAL 1

1.1= 8

1.2

• 2³ . 3 7,5
• :3(3 + 4) = 21
• :81
• :16

1.7 :125

2.1 :3 + 2 + 4 = 9

9 ÷ 3 = 3 Yes!

2.2: 324 = 2² x 3 4

2.3:= (2² x 3 4 )

= 2 x 3²

= 18

2.4: Yes! 18 x 18 = 324 /18² = 324

• :9
• : $\frac{6}{2}$ = 3

3.3: $\sqrt{9+\text{16}}$ = $\sqrt{\text{25}}$ = 5

3.4: 4 ${x}^{8}$

• :4 3 = 64
• :3

Enrichment exercise

1. d

2. c

3. d

4. $\frac{\text{180}\left(\text{10}-2\right)}{\text{10}}$ = 144º (one angle) (1 440 – 240) ÷ 8 = 150 ( d )

5. b 3 1992 ends on 1

6. d AB = 6

(2 x ) 2 + x 2 = 36

4 x 2 + x 2 = 36

5 x 2 = 36

TEST 1

• :23, 29
• :1, 2, 3, 6, 12
• :4, 6, 12

2. :* 2 1 + 2 + 1 + 3 + 1 + 5 + 6 + 3 = 22

3.1 $\sqrt{\text{100}}$ = 10

3.2 2 3 = 8

3.3 $\sqrt{\frac{\text{25}}{9}}$ = $\frac{5}{3}$ = 1 $\frac{2}{3}$

3.4 $\sqrt{\frac{4}{\text{100}}}$ = $\frac{2}{\text{10}}$ = 0,2 / $\frac{1}{5}$

3.5 $\sqrt{\text{64}}$ = 8

• :2 x 3 = 6
• :9
• :4 – 1 = 3

4. $\sqrt[3]{{2}^{6}{\mathrm{x3}}^{3}}$ = 2 2 x 3

= 4 x 3

= 12

5. (2) = 2 2 = 4

(4) = 4 4 = 256

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