Prime factors, square roots and cube rootsm31099Prime factors, square roots and cube roots1.12009/08/07 13:25:08.406 GMT-52009/08/07 13:32:50.495 GMT-5gertbezuidenhoutgert bezuidenhoutgertb@mweb.co.zagertbezuidenhoutgert bezuidenhoutgertb@mweb.co.zagertbezuidenhoutgert bezuidenhoutgertb@mweb.co.zaMathematics and StatisticsenMATHEMATICSGrade 8 THE NUMBER SYSTEMModule 2PRIME FACTORS, SQUARE ROOTS AND CUBE ROOTSCLASS ASSIGNMENT 11. Prime factorsHow do you write a number as the product of its prime factors? And how do you write it in exponent notation? E.g. Question: Write 24 as the product of its prime factors(remember that prime factors are used as divisors only)
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Prime factors of 24 = {2; 3} 24 as product of its prime factors: 24 = 2 x 2 x 2 x 3 24 = 23 x 3 (exponential notation)Now express each of the following as the product of their prime factors(exponential notation) and also write the prime factors of each.2. Square roots and cube rootsHow do you determine the square root (
size 12{ sqrt {} } {})or cube root (
3 size 12{ nroot { size 8{3} } {} } {})of a number with the help of prime factors?Do you recall this? Determine:
324 size 12{ sqrt {"324"} } {}Step 1: break down into prime factors Step 2: write as product of prime factors (in exponential notation)Step 3:
324 size 12{ sqrt {"324"} } {} means (324)½ (obtain half of each exponent)
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Therefore:
324 size 12{ sqrt {"324"} } {} = (22 x 34)½ = 21 x 32 = 2 x 9 = 18(324 is a perfect square, because 18 x 18 = 324)Remember:
size 12{ sqrt {```} } {} means (......)½ and
3 size 12{ nroot { size 8{3} } {```} } {} means (......)1/38x123=2x12÷3=4 size 12{ nroot { size 8{3} } {8x rSup { size 8{"12"} } } `=`2x rSup { size 8{"12" div 3=4} } } {} therefore
2x4 size 12{2x rSup { size 8{4} } } {}2.1 Calculate with the help of prime factors:(i)
1 024 size 12{ sqrt {"1 024"} } {}
2. Determine the answers without using a calculator.2.1
3.3.3.3.323 size 12{ nroot { size 8{3} } {3 "." 3 "." 3 "." 3 "." 3 rSup { size 8{2} } } } {} = 2.2
53a6b153 size 12{ nroot { size 8{3} } {5 rSup { size 8{3} } `a rSup { size 8{6} } `b rSup { size 8{"15"} } } } {} = 2.3
8÷125×273 size 12{ nroot { size 8{3} } {8` div `"125"` times `"27"} } {} = 2.4
643+(643)3 size 12{ nroot { size 8{3} } {"64"} `+` \( ` nroot { size 8{3} } {"64"} ` \) rSup { size 8{3} } } {} = 2.5
2(83)3 size 12{2` \( ` nroot { size 8{3} } {8} ` \) rSup { size 8{3} } } {} = 2.6
169 size 12{ sqrt {"169"} } {} = 2.7
(6+4×12)2 size 12{ sqrt { \( 6`+`4` times `"12" \) rSup { size 8{2} } } } {} = 2.8
6×18×12 size 12{ sqrt {6` times 1`8` times `"12"} } {} = 2.9
2(9)2 size 12{2 \( sqrt {9} \) rSup { size 8{2} } } {} = 2.10
(6+3)2−33 size 12{ sqrt { \( 6`+`3 \) rSup { size 8{2} } } ` - `3 rSup { size 8{3} } } {} = CLASS ASSIGNMENT 21. Give the meaning of the following in your own words (discuss it in your group)LCM:Explain it with the help of an exampleBCD:Explain it with the help of an example2. How would you determine the LCM and BCD of the following numbers?8; 12; 20Step 1: Write each number as the product of its prime factors.(Preferably not in exponential notation) 8 = 2 x 2 x 2 12 = 2 x 2 x 3 20 = 2 x 2 x 5Step 2: First determine the BCD (the number/s occurring in each of the three)Suggestion: If the 2 occurs in each of the three, circle the 2 in each number and write it down once), etc. BCD = 2 x 2 = 4Step 3: Now determine the LCM. First write down the BCD and then find the number that occurs in two of the numbers and write it down, finally writing what is left over) LCM = 4 x 2 x 3 x 5 = 1203. Do the same and determine the BCD and LCM of the following:38; 57; 95Calculate it here: 38 = .................................................................... 57 = .................................................................... 95 = ....................................................................BCD = .................................. and LCM = ..................................Assessment
Assessment of myself:by myself:Assessment by Teacher:I can…1234Critical Outcomes1234determine prime factors of a number; (Lo 1.2.6)Critical and creative thinkingexpress a number as the product of its prime factors; (Lo 1.2.6; 1.2.3)Collaboratingexpress prime factors in exponent notation; (Lo 1.2.7)Organising en managingdetermine the square root of a number; (Lo 1.2.7)Processing of informationdetermine the cube root of a number. (Lo 1.2.7)Communicationdetermine/define the smallest common factor (LCM); (Lo 1.2.6)Problem solvingdetermine/define the biggest common divider (BCD). (Lo 1.2.6)Independence
goodaveragenot so good
Comments by the learner:My plan of action:My marks:I am very satisfied with the standard of my work.< Date:I am satisfied with the steady progress I have made.Out of:I have worked hard, but my achievement is not satisfactory.Learner:I did not give my best. >
Comments by parents:Comments by teacher:Signature: Date: Signature: Date:
Tutorial 1: (Number Systems)Total: 301. Simplify:1.1
100−36 size 12{ sqrt {"100"` - `"36"} } {} [1]1.2
2549 size 12{ sqrt { { {"25"} over {"49"} } } } {} [1]1.3
26315 size 12{ sqrt {2 rSup { size 8{6} } 3 rSup { size 8{"15"} } } } {} [2]1.4
9(9+16) size 12{ sqrt {9} ` \( sqrt {9} `+` sqrt {"16"} \) } {} [3]1.5 9² [1]1.6
a=4,a= size 12{ sqrt {a} `=`"4,"~a`=`} {} [1]1.7
a3=5,a= size 12{ nroot { size 8{3} } {a} `=`"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
324 size 12{ sqrt {"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
81 size 12{ sqrt {"81"} } {} [1]3.2
364 size 12{ sqrt { { {"36"} over {4} } } } {} [2]3.3
32+42 size 12{ sqrt {3 rSup { size 8{2} } `+`4 rSup { size 8{2} } } } {} [2]3.4
16x16 size 12{ sqrt {"16"x rSup { size 8{"16"} } } } {} [2]4. If x = 3, determine:4.1
4x size 12{4 rSup { size 8{x} } } {} [2]4.2
27x size 12{ nroot { size 8{x} } {"27"} } {} [2] [11]Tutorial
I demonstrate knowledge and understanding of:Learning outcomes00000000001.natural numbers (N) and whole numbers (N0)1.12.the identification of the different types of numbers;1.13.compound numbers;1.2.64.divisibility rules;1.2.65.the multiples of a number;1.2.66.the factors of a number;1.2.67.prime numbers;1.18.prime factors;1.2.69.expressing a number as the product of its prime factors;1.2.6; 1.2.310.expressing prime factors in exponent notation;1.2.311.even and odd numbers;1.112.square roots of a number;1.2.713.cube roots of a number;1.2.714.the smallest common factor (LCM);1.2.615.the biggest common divider (BCD).1.2.6
The learner’s …1234work 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.
My BEST marks:Comments by teacher:Date:Out of:Learner:Signature: Date:
Parent signature: Date: Test 1: (Number Systems)Total: 301. 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
36+64 size 12{ sqrt {"36"`+`"64"} } {} [2] 3.2
293 size 12{ nroot { size 8{3} } {2 rSup { size 8{9} } } } {} [2]3.3
279 size 12{ sqrt {2 { { size 8{7} } over { size 8{9} } } } } {} [3] 3.4
0,04 size 12{ sqrt {"0,04"} } {} [2]3.5
100− 36 size 12{ sqrt {"100"`-`" 36"} } {} [2] 3.6
8×273 size 12{ nroot { size 8{3} } {8 times "27"} } {} [2]3.7
(9)2 size 12{ \( sqrt {9} \) rSup { size 8{2} } } {} [2] 3.8
64−13 size 12{ nroot { size 8{3} } {"64"` - `1} } {} [2] [17]4. Determine
1 7283 size 12{ nroot { size 8{3} } {"1 728"} } {} using prime factors, without using a calculator. [5]5. Bonus questionIf (n) means nn what is the value of ((2)) ? [2]Enrichment Exercise for the quick learner(Learning unit1)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) n²p + 2 c) n+p+1 d) 2n+3p+5 e) 2n+p2. 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) R403. How many triangles are there in the figure?a) 8 b) 12 c) 14 d) 16 e) 204. 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 31993 is ....a) 1 b) 3 c) 6 d) 7 e) 96. The figure below has 5 squares. If AB = 6, the area of the figure is...a) 12 b) 20 c) 24 d) 36 e) impossibleAssessment
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.