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To differentiate between rational and irrational numbers

Module by: Siyavula Uploaders. E-mail the author

MATHEMATICS

Grade 8

RATIONAL NUMBERS, CIRCLES AND TRIANGLES

Module 15

DIFFERENTIATING BETWEEN RATIONAL AND IRRATIONAL NUMBERS

ACTIVITY 1

Differentiating between rational and irrational numbers

[LO 1.2.7]

1. Can you remember what each of the following represents?

N = { ........................................................................... }

N0= { ........................................................................... }

Z = { ........................................................................... }

R = { ........................................................................... }

2. Provide the definition for:

a rational number:

an irrational number:

3. How would you represent each of the following?

3.1 Rational number......................... 3.2 Irrational number .........................

4. Complete the following table by marking relevant numbers with an X:

Figure 1
Figure 1 (Picture 435.png)

5. Select the required numbers from the list:

2323 size 12{ { { size 8{ - 2} } over { size 8{3} } } } {}; 1 + 44 size 12{ sqrt {4} } {} ; 9+49+4 size 12{ sqrt {9+4} } {} ; -4 ; 12151215 size 12{"12" { { size 8{1} } over { size 8{5} } } } {} ; 1+221+22 size 12{ { {1+ sqrt {2} } over { sqrt {2} } } } {}

5.1 Integers:

5.2 Rational numbers:

5.3 Irrational numbers:

6. Explain what you know about an equivalent fraction.

7. Provide two equivalent fractions for the following: 2727 size 12{ { { size 8{2} } over { size 8{7} } } } {} = ............... = ...............

8. Provide the terms used to identify each of the following (e.g. proper fraction):

8.1 2727 size 12{ { { size 8{2} } over { size 8{7} } } } {}

8.2 7272 size 12{ { { size 8{7} } over { size 8{2} } } } {}

8.3 627627 size 12{6 { { size 8{2} } over { size 8{7} } } } {}

8.4 0,67

8.5 0,6˙7˙0,6˙7˙ size 12{0, { dot {6}} { dot {7}}} {}

8.6 23 %

Any of the above can be reduced to any of the others.

ACTIVITY 2:

Reduction of fractions to decimal numbers / recurring decimal numbers and vice versa

[LO 1.2.2, 1.2.6, 1.3, 1.6.1, 1.9.1]

  1. Use your pocket calculator to reduce the following fraction to a decimal number:
Figure 2
Figure 2 (Picture 443.png)

2. Explain how you would reduce this to a decimal number without the use of your pocket calculator. There are two methods:

Method 1: .................................................. (reduce denominator to 10 / 100 / 1 000)

Method 2: .................................................. (do division)

(Let your educator assist you.)

  • Do you see that the answer is the same – if the denominator cannot be reduced to multiples of 10 you have to apply the second method.

3. Now reduce each of the following to decimal numbers (round off, if necessary, to two digits):

3.1 5858 size 12{ { { size 8{5} } over { size 8{8} } } } {} ..................................................

3.2 134134 size 12{ { { size 8{"13"} } over { size 8{4} } } } {} ..................................................

3.3 534534 size 12{5 { { size 8{3} } over { size 8{4} } } } {} ..................................................

3.4 378378 size 12{3 { { size 8{7} } over { size 8{8} } } } {} ..................................................

3.5 6767 size 12{ { { size 8{6} } over { size 8{7} } } } {} ..................................................

3.6 7979 size 12{ { { size 8{7} } over { size 8{9} } } } {} ..................................................

4. Write the following decimal numbers as fractions or mixed numbers:(N.B.: All fractions have to be presented in their simplest form.)

4.1 6,008 ..................................................

4.2 4,65 ..................................................

4.3 0,375 ..................................................

4.4 7,075 ..................................................

4.5 13,65 ..................................................

4.6 0,125 ..................................................

5. How do we reduce fractions to recurring decimal numbers?

E.g. 511511 size 12{ { { size 8{5} } over { size 8{"11"} } } } {}

Step 1: place a comma after the 5, i.e. 5, 0000

Step 2: carry on dividing until a pattern becomes visible - the pattern will be indicated by the recurring numbers.

Figure 3
Figure 3 (Picture 445.png)

Now try the following:

5.1 7979 size 12{ { { size 8{7} } over { size 8{9} } } } {}

5.2 556556 size 12{ - 5 { { size 8{5} } over { size 8{6} } } } {}

5.3 3139931399 size 12{3 { { size 8{"13"} } over { size 8{"99"} } } } {}

6. What is noticeable about fractions that are recurring decimal numbers (with regard to the denominator)?

7. Now, before we provide the steps for reducing a recurring decimal number to a common fraction, see if you are able to write the following as fractions by making use of the information from no. 6.

Figure 4
Figure 4 (Picture 449.png)

8. The following provides complete steps for reducing a recurring decimal number to a common fraction:

Figure 5
Figure 5 (Picture 454.png)

Suggestion: Multiply by 10 (if you have one recurring figure). Multiply by 100 (if there are 2 recurring figures), etc.

9. Now try to do no. 7.2 in the way that is discussed in no. 8.

ACTIVITY 3

Reducing percentages to fractions and vice versa

[LO 1.2.2, 1.2.6, 1.6.1, 1.9.1]

1. What is the meaning of % (percentage)? .....................................................................

2. If you have to reduce any fraction to a percentage, you have to reduce the denominator to 100.

  • If this is not possible, you have to x
    Figure 6
    Figure 6 (Picture 459.png)
    (This principle can be applied in any situation, e.g. when you want to reduce a test that is marked out of 15 to a mark out of 50, you need to multiply by 501501 size 12{ { { size 8{"50"} } over { size 8{1} } } } {})

Reduce the following mathematics test marks from a grade 8 class to percentages (to one decimal figure, where necessary):

2.1 17201720 size 12{ { { size 8{"17"} } over { size 8{"20"} } } } {} .......................................

2.2 19401940 size 12{ { { size 8{"19"} } over { size 8{"40"} } } } {} .......................................

2.3 38503850 size 12{ { { size 8{"38"} } over { size 8{"50"} } } } {} .......................................

2.4 45604560 size 12{ { { size 8{"45"} } over { size 8{"60"} } } } {} .......................................

3. Reduce each of the following percentages to a common fraction (or a mixed number):

3.1 55 % .......................................

3.2 15,5% .......................................

3.3 16 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}% .......................................

3.4 623623 size 12{6 { { size 8{2} } over { size 8{3} } } } {}% .......................................

4. Each South African citizen should have access to some means of transport.

Bolokanang has a community of 25 500 people. Study the accompanying table indicating the number of people that use the given means of transport and answer the questions that follow.

Table 1
Vehicle Number of users
Bicycle 418418 size 12{4 { { size 8{1} } over { size 8{8} } } } {}%
Car 3 5 3 5 size 12{ { { size 8{3} } over { size 8{5} } } } {}
Motorbike 0,085

4.1 Indicate how many inhabitants make use of:

a) a bicycle

b) a car

c) a motorbike

4.2 Express the number of inhabitants that use a car as a fraction of those who travel by bicycle.

4.3 Which percentage of the inhabitants has no vehicle?

4.4 Which other means of transport do farm labourers use to get to the nearest town?

4.5 If the number of job opportunities in rural areas should increase, the fraction of citizens who use cars for transport will double. What fraction of the community will be using cars for transport under such conditions?

ACTIVITY 4

Adding and subtracting rational numbers (fractions)

[LO 1.2.2, 1.2.5, 1.2.6, 1.6.2, 1.7.1, 1.7.2, 1.9.1]

1. Reduce each of the following compound numbers to improper fractions.This is very important in addition, subtraction, multiplication and division of fractions.

1.1 5 4747 size 12{ { { size 8{4} } over { size 8{7} } } } {} ................................ 1.2 7 7979 size 12{ { { size 8{7} } over { size 8{9} } } } {} ................................

2. What is of cardinal importance before attempting to add or subtract fractions?

3. Show whether you are able to do the following:

3.1 8 - 4 3737 size 12{ { { size 8{3} } over { size 8{7} } } } {}

3.2 3 1919 size 12{ { { size 8{1} } over { size 8{9} } } } {} - 1 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}

  • Note this: The denominators must be similar when you add fractions together or subtract them from one another.

e.g. 2 4747 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 1 6767 size 12{ { { size 8{6} } over { size 8{7} } } } {}

2 – 1 = 1 and

4747 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 6767 size 12{ { { size 8{6} } over { size 8{7} } } } {}

( 4 – 6 --- this is not possible. Carry one whole: 1 = 7777 size 12{ { { size 8{7} } over { size 8{7} } } } {})

( 4 + 7 = 11 --- yes, 11 – 6 = 5)

Answer: 5757 size 12{ { { size 8{5} } over { size 8{7} } } } {}

  • You could also reduce compound numbers to improper fractions and make the denominators similar.
  • e.g.. 187137=57187137=57 size 12{ { { size 8{"18"} } over { size 8{7} } } - { { size 8{"13"} } over { size 8{7} } } = { { size 8{5} } over { size 8{7} } } } {} (18 – 13 = 5: The denominators are the same. Subtract one numerator from the other.)

4. Do the following:

4.1 4 1717 size 12{ { { size 8{1} } over { size 8{7} } } } {} + 4 16421642 size 12{ { { size 8{"16"} } over { size 8{"42"} } } } {}

4.2 36 - 15 611611 size 12{ { { size 8{6} } over { size 8{"11"} } } } {}

4.3 18+0,6253818+0,62538 size 12{ { { size 8{1} } over { size 8{8} } } +0,"625" - { { size 8{3} } over { size 8{8} } } } {}

4.4 4510+712+6344510+712+634 size 12{4 { { size 8{5} } over { size 8{"10"} } } +7 { { size 8{1} } over { size 8{2} } } +6 { { size 8{3} } over { size 8{4} } } } {}

4.5 7 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {} - 4 7878 size 12{ { { size 8{7} } over { size 8{8} } } } {}

4.6 7a - a4a4 size 12{ { { size 8{a} } over { size 8{4} } } } {}a/4

4.7 9a+6ab3b9a+6ab3b size 12{ { { size 8{9} } over { size 8{a} } } + left ( { { size 8{6} } over { size 8{ ital "ab"} } } - { { size 8{3} } over { size 8{b} } } right )} {}

4.8 - 6 + 2 6767 size 12{ { { size 8{6} } over { size 8{7} } } } {}

4.9 5 - (4 4949 size 12{ { { size 8{4} } over { size 8{9} } } } {} + 2 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {})

4.10 3 1313 size 12{ { { size 8{1} } over { size 8{3} } } } {}a- 2 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {}a

ACTIVITY 1.5

Multiplication and division of rational numbers (fractions)

[LO 1.2.6, 1.6.2]

  • You did this in grade 7 – let's refresh the memory.

1. Multiplication:

  • Important: Write all compound numbers as fractions.Then do crosswise cancellation.

Try the following:

  • 1 1414 size 12{ { { size 8{1} } over { size 8{4} } } } {} × 2 1212 size 12{ { { size 8{1} } over { size 8{2} } } } {} × 4

2. Division:

  • The reciprocal plays an important role in the division of fractions.

Use an example to explain this term.

e.g. 13÷2313÷23 size 12{ { { size 8{1} } over { size 8{3} } } div { { size 8{2} } over { size 8{3} } } } {}

  • Both numbers are fractions
  • Change ÷ to the × sign and obtain the reciprocal of the denominator (fraction following the ÷ sign).
  • Do cancellation as with multiplication.

3. Do the following:

3.1 8 ÷ 811811 size 12{ { { size 8{8} } over { size 8{"11"} } } } {}

3.2 18 ÷ 7878 size 12{ { { size 8{7} } over { size 8{8} } } } {}

3.3 56÷5256÷52 size 12{ { { size 8{5} } over { size 8{6} } } div { { size 8{5} } over { size 8{2} } } } {}

3.4 -2 2323 size 12{ { { size 8{2} } over { size 8{3} } } } {} ÷ -1 7979 size 12{ { { size 8{7} } over { size 8{9} } } } {}

3.5 6 3434 size 12{ { { size 8{3} } over { size 8{4} } } } {}mn ÷ -6 m3

3.6 4xy3ab÷2x3a4xy3ab÷2x3a size 12{ { { size 8{ - 4 ital "xy"} } over { size 8{3 ital "ab"} } } div { { size 8{ - 2x} } over { size 8{3a} } } } {}-

Assessment

Table 2
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:
1.2 recognises, classifies an represents the following numbers to describe and compare them:
1.2.2 decimals, fractions and percentages;
1.2.5 additive and multiplicative inverses;
1.2.6 multiples and factors;
1.2.7 irrational numbers in the context of measure­ment (e.g. ππ size 12{π} {}and square and cube roots of non-perfect squares and cubes);
1.3 recognises and uses equivalent forms of the rational numbers listed above;
1.6 estimates and calculates by selecting and using operations appropriate to solving problems that involve:
1.6.1 rounding off;
1.6.2 multiple operations with rational numbers (including division with fractions and decimals);
1.7 uses a range of techniques to perform calculations, including:
1.7.1 using the commutative, associative and distributive properties with rational numbers;
1.7.2 using a calculator;
1.9 recognises, describes and uses:
1.9.1 algorithms for finding equivalent fractions;
1.9.2 the commutative, associative and distributive properties with rational numbers (the expecta­tion is that learners should be able to use these properties and not necessarily to know the names of the properties).

Memorandum

ACTIVITY 1

1. Natural numbers

Counting numbers

Integers

Real numbers

2. abab size 12{ { {a} over {b} } } {}; b ≠ 0

2 2 size 12{ sqrt {2} } {}
(1)

3.1 Q

  • Q 1

4.

Table 3
  size 12{ {2} wideslash {7} } {} 0 1 1 size 12{ sqrt {1} } {} 3 3 size 12{ sqrt {3} } {} 9 3 9 3 size 12{ nroot { size 8{3} } {9} } {} 8 3 8 3 size 12{ nroot { size 8{3} } {8} } {} 2,47 1, 45 1, 45 size 12{ sqrt {1,"45"} } {} size 12{ sqrt { {4} wideslash {8} } } {} size 12{ sqrt { {"16"} wideslash { sqrt {9} } } } {}
Rational        
Irrational            
  • 1 + 44 size 12{ sqrt {4} } {}; -4
  • 2323 size 12{ { { - 2} over {3} } } {}; 12 1515 size 12{ { {1} over {5} } } {}
  • 9+49+4 size 12{ sqrt {9+4} } {}; 1+221+22 size 12{ { {1+ sqrt {2} } over { sqrt {2} } } } {}

6. Equal in value

7. 414414 size 12{ { {4} over {"14"} } } {} = 624624 size 12{ { {6} over {"24"} } } {} etc

  • Proper fraction
  • Inproper fraction
  • Mixed number
  • Decimal number
  • Recurring decimal number
  • Percentage

ACTIVITY 2

1. 2,15

  • 0,625
  • 3,25
  • 5,75
  • 2,875
  • 6,00076,0007 size 12{ { {6,"000"} over {7} } } {} = 0,8571 . . . ≈ 0,86
  • 7,00097,0009 size 12{ { {7,"000"} over {9} } } {} = 0,777 . . . = 0, 77 size 12{ {7} cSup { size 8{ cdot } } } {} or 0,8
  • 6 8100081000 size 12{ { {8} over {"1000"} } } {} = 6 11251125 size 12{ { {1} over {"125"} } } {}
  • 4 6510065100 size 12{ { {"65"} over {"100"} } } {} = 4 13201320 size 12{ { {"13"} over {"20"} } } {}
  • 37510003751000 size 12{ { {"375"} over {"1000"} } } {} = 3838 size 12{ { {3} over {8} } } {}
  • 7 751000751000 size 12{ { {"75"} over {"1000"} } } {} = 7 340340 size 12{ { {3} over {"40"} } } {}
  • 13 6510065100 size 12{ { {"65"} over {"100"} } } {} = 13 13201320 size 12{ { {"13"} over {"20"} } } {}
  • 12510001251000 size 12{ { {"125"} over {"1000"} } } {} = 1818 size 12{ { {1} over {8} } } {}

5.1 7,00097,0009 size 12{ { {7,"000"} over {9} } } {} = 0, 77 size 12{ {7} cSup { size 8{ cdot } } } {}

5.2 -5,8 33 size 12{ {3} cSup { size 8{ cdot } } } {}5,00065,0006 size 12{ { {5,"000"} over {6} } } {} = 0,8333 . . .

5.3 3, 11 size 12{ {1} cSup { size 8{ cdot } } } {}33 size 12{ {3} cSup { size 8{ cdot } } } {}13,00009913,000099 size 12{ { {"13","0000"} over {"99"} } } {} = 0,1313 . . .

7.1 3939 size 12{ { {3} over {9} } } {} = 1313 size 12{ { {1} over {3} } } {}

7.2 45994599 size 12{ { {"45"} over {"99"} } } {} = 511511 size 12{ { {5} over {"11"} } } {}

7.3 2399023990 size 12{ { {"23"} over {"990"} } } {}

7.4 39003900 size 12{ { {3} over {"900"} } } {} = 13001300 size 12{ { {1} over {"300"} } } {}

9. 0, 44 size 12{ {4} cSup { size 8{ cdot } } } {}55 size 12{ {5} cSup { size 8{ cdot } } } {} = x

x = 0,4545 . . . 

100 x = 45,4545 . . .

  • –  99 x = 45

x = 45994599 size 12{ { {"45"} over {"99"} } } {} = 511511 size 12{ { {5} over {"11"} } } {}

ACTIVITY 3

2.1 17x520x517x520x5 size 12{ { {"17"x5} over {"20"x5} } } {} = 85%

2.2 19401940 size 12{ { {"19"} over {"40"} } } {} x 10011001 size 12{ { {"100"%} over {1} } } {} = 47,5%

2.3 38x250x238x250x2 size 12{ { {"38"x2} over {"50"x2} } } {} = 76%

2.4 45604560 size 12{ { {"45"} over {"60"} } } {} x 10011001 size 12{ { {"100"%} over {1} } } {} = 75%

3.1 5510055100 size 12{ { {"55"} over {"100"} } } {} = 11201120 size 12{ { {"11"} over {"20"} } } {}

3.2 15,510015,5100 size 12{ { {"15",5} over {"100"} } } {} = 0,155 = 15510001551000 size 12{ { {"155"} over {"1000"} } } {} = 3120031200 size 12{ { {"31"} over {"200"} } } {}

3.3 3320033200 size 12{ { {"33"} over {"200"} } } {}

3.4 2030 {02030 {0 size 12{ { {2 { {0}}} over {"30 {"{0}}} } } {} = 230230 size 12{ { {2} over {"30"} } } {}

4.a) 3380033800 size 12{ { {"33"} over {"800"} } } {} x 255001255001 size 12{ { {"25500"} over {1} } } {} size 12{ approx } {} 1 052

b) 3535 size 12{ { {3} over {5} } } {} x 255001255001 size 12{ { {"25500"} over {1} } } {} = 15 300

c) 851000851000 size 12{ { {"85"} over {"1000"} } } {} x 255001255001 size 12{ { {"25500"} over {1} } } {} = 2 167,5 size 12{ approx } {} 2 168

  • (14,5) 153001052153001052 size 12{ { {"15300"} over {"1052"} } } {} = 76505267650526 size 12{ { {"7650"} over {"526"} } } {} = 38252633825263 size 12{ { {"3825"} over {"263"} } } {}
  • 25 500 – 18 520 = 6 980

4.4

4.5 3535 size 12{ { {3} over {5} } } {} x 2121 size 12{ { {2} over {1} } } {} = 6565 size 12{ { {6} over {5} } } {} = 115115 size 12{1 { {1} over {5} } } {}

ACTIVITY 4

1.1 397397 size 12{ { {"39"} over {7} } } {}

1.2 709709 size 12{ { {"70"} over {9} } } {}

2. Numbers must be the same

3.1 347347 size 12{3 { {4} over {7} } } {}

3.2 2291822918 size 12{2 { {2 - 9} over {"18"} } } {} = 120918120918 size 12{1 { {"20" - 9} over {"18"} } } {} = 1111811118 size 12{1 { {"11"} over {"18"} } } {}

4.1 297297 size 12{ { {"29"} over {7} } } {} + 1844218442 size 12{ { {"184"} over {"42"} } } {} = 174+18442174+18442 size 12{ { {"174"+"184"} over {"42"} } } {} = 3584235842 size 12{ { {"358"} over {"42"} } } {} = 8224282242 size 12{8 { {"22"} over {"42"} } } {} = 8112181121 size 12{8 { {"11"} over {"21"} } } {}

4.2 21 - 611611 size 12{ { {6} over {"11"} } } {} = 2051120511 size 12{"20" { {5} over {"11"} } } {}

  • 0,125 + 0,625 – 0,375 = 0,375
  • 1710+10+15201710+10+1520 size 12{"17" { {"10"+"10"+"15"} over {"20"} } } {} = 173520173520 size 12{"17" { {"35"} over {"20"} } } {} = 181520181520 size 12{"18" { {"15"} over {"20"} } } {} = 18341834 size 12{"18" { {3} over {4} } } {}
  • 332124332124 size 12{3 { {3 - "21"} over {"24"} } } {} = 2112421124 size 12{2 { {"11"} over {"24"} } } {}
  • {} 28 a 2 a 4 28 a 2 a 4 size 12{ { {"28"`a rSup { size 8{2} } - a} over {4} } } {}

4.7 + (63aab)(63aab) size 12{\( { {6 - 3a} over { ital "ab"} } \)} {} = 9b+63aab9b+63aab size 12{ { {9b+6 - 3a} over { ital "ab"} } } {}

4.8 6161 size 12{ { { - 6} over {1} } } {} + 207207 size 12{ { {"20"} over {7} } } {} = 42+20742+207 size 12{ { { - "42"+"20"} over {7} } } {} = 227227 size 12{ { { - "22"} over {7} } } {} = 317317 size 12{ - 3 { {1} over {7} } } {}

  • 5 – 64+6964+69 size 12{ left (6 { {4+6} over {9} } right )} {} = 5 – 61096109 size 12{6 { {"10"} over {9} } } {} = 5 – 719719 size 12{7 { {1} over {9} } } {}

= – 649649 size 12{ { {"64"} over {9} } } {}

= 4564945649 size 12{ { {"45" - "64"} over {9} } } {}

= 199199 size 12{ { { - "19"} over {9} } } {} = 219219 size 12{ - 2 { {1} over {9} } } {}

  • 10a310a3 size 12{ { {"10"a} over {3} } } {}5a25a2 size 12{ { {5a} over {2} } } {} = 20a15a620a15a6 size 12{ { {"20"a - "15"a} over {6} } } {}

= 5a65a6 size 12{ { {5a} over {6} } } {}

ACTIVITY 5

1. 514514 size 12{ { {5} over { {} rSub { size 8{1} } { {4}}} } } {} x 5252 size 12{ { {5} over {2} } } {} x 411411 size 12{ { { { {4}} rSup { size 8{1} } } over {1} } } {} = 252252 size 12{ { {"25"} over {2} } } {} = 12121212 size 12{"12" { {1} over {2} } } {}

3.1 8181 size 12{ { {8} over {1} } } {} ÷ 811811 size 12{ { {8} over {"11"} } } {} = 811811 size 12{ { { { {8}} rSup { size 8{1} } } over {1} } } {} x 11811181 size 12{ { {"11"} over { { {8}} rSub { size 8{1} } } } } {} = 11

3.2 181181 size 12{ { {"18"} over {1} } } {} x 8787 size 12{ { {8} over {7} } } {} = 14471447 size 12{ { {"144"} over {7} } } {} = 20472047 size 12{"20" { {4} over {7} } } {}

3.3 51635163 size 12{ { { { {5}} rSup { size 8{1} } } over { { {6}} rSub { size 8{3} } } } } {} x 21512151 size 12{ { { { {2}} rSup { size 8{1} } } over { { {5}} rSub { size 8{1} } } } } {} = 1313 size 12{ { {1} over {3} } } {}

3.4 81318131 size 12{ { { - { {8}} rSup { size 8{1} } } over { { {3}}"" lSub { size 8{1} } } } } {} x 9316293162 size 12{ { { - { {9}} rSup { size 8{3} } } over { { {1}} { {6}} rSub { size 8{2} } } } } {} = 3232 size 12{ { {3} over {2} } } {} = 112112 size 12{1 { {1} over {2} } } {}

3.5 279mn4279mn4 size 12{ { { { {2}} { {7}} rSup { size 8{9} } ital "mn"} over {4} } } {} x 162m3162m3 size 12{ { {1} over { - { {6}}"" lSub { size 8{2} } m rSup { size 8{3} } } } } {} = 9n8m29n8m2 size 12{ { { - 9n} over {8m rSup { size 8{2} } } } } {}

3.6 42xy31ab42xy31ab size 12{ { { - { {4}} rSup { size 8{2} } ital "xy"} over { { {3}}"" lSub { size 8{1} } { {a}}b} } } {} x 3a2x3a2x size 12{ { { { {3}} { {a}}} over { - { {2}} { {x}}} } } {} = 2yb2yb size 12{ { {2y} over {b} } } {}

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A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

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