(1, 2, 3 . . .) (−1,−2,−3 . . .)0, 3, 76,−2,−11, 995, . . .

2 −5

2 −52 −5 >

2 > −5

72

5

25 9

9 5 711 7 3

= 5 = 7 + (5 · 2) = 17

= 4 = 7 + (3 · 2) = 13

= 3 = 7 + (1 · 2) = 9

= 2 = 7− (1 · 2) = 5

= 1 = 7− (3 · 2) = 1

= 5 = 7− (5 · 2) = −3

a

b ab

b

b

a12

12

34

34

910

910

24

12

(

2

4

)

=

(

�2 · 1�2 · 2 · 1

) (

2

4

)

=

(

1

2

)

(

37

)

(

93

)

(

5060

)

3 7

37

9 = 3 · 3 3(

93

)

=(

�3·3·1�3·1

)

. 9 > 3

93

31 3

50 = 5 · 5 · 2 60 = 5 · 3 · 2 · 2 5060 =

(

�5·5·�2·1�5·5·�2·2·1

)

. 50 < 60

5060

56

12 , 3

4 , 23

34

12

1

2<

2

3<

3

4

37

49

7 9 6363 9 7 = 63

7 63

(

3

7

)

=9

9

(

3

7

)

=

(

27

63

)

99 1 27

6337

9 63

(

4

9

)

=7

7

(

4

9

)

=

(

28

63

)

2863

49

6328 63 4

937

2863

2763

49

37

3 + (−3)5 + (−5)(−11.5) + (11.5)37 + −3

73 (−3)

5 (−5) (−11.5) 11.5 37

37

0

19.6−4

9x

xy2

(x− 3)

−1

19.6 −19.6

−49

49

x −x

xy2 −xy2

(x− 3) −(x− 3) = 3− x

−1(x− 3) (x + 3)

7 7 −7 77 −7 7

−7 | − 7|

|x| x

|5 + 4|3− |4− 9|| − 5− 11|−|7− 22|

|5 + 4| = |9|= 9

3− |4− 9| = 3− | − 5|= 3− 5

= −2

| − 5− 11| = | − 16|= 16

−|7− 22| = −| − 15|= −(15)

= −15

(1, 2, 3 . . .)(−1,−2,−3 . . .)

ab

35

−1

a, b, c, d e

12 , 1

3 , 14

1112 , 12

11 , 1310

3960 , 49

80 , 59100

711 , 8

13 , 1219

2244

927

1218

315420

1.001

(5− 11)

(x + y)

(x− y)

11− | − 4||4− 9| − | − 5|| − 5− 11|7− |22− 15− 19|−| − 7|| − 2− 88| − |88 + 2|

a = −3 b = 3 c = 9 d = 12 e = 15

a = 13 b = 7

12 c = 2235

14 < 1

3 < 12

1112 < 12

11 < 1310

59100 < 49

80 < 3960

813 < 12

19 < 711

12132334

−1.001

6− (x + y)

(y − x)

7

0

16

−5

−7

0

2 + 32 3 5

2 + 3 = 5

−2 + 3−2 3 +1

−2 + 3 = 1

2− 3

23 −1

2− 3 = −1

13

14

112

=1

3− 1

4=

1

12.

13 + 1

4

35 + 1

6

5 6 5 6

5 6 30 5 6 3030

3035 3

5 30 618 18 = 3× 6

35

1830

16 30

5 516

530

18 5 23 2330

3

5+

1

6=

18

30+

5

30=

23

30

ac

+ bc

= a+bc

1411 + 1

999 9

11 99 1411

19 99

11 99 1411

14·911·9 = 126

99

9 9/9 = 1 1

9 99 19

1·119·11 = 11

99

14

11+

1

9=

126

99+

11

99=

137

99

112 + 2

936 12 9 36

3612 35 1

121·312·3 = 3

369 36 2

92·49·4 = 8

36

1

12+

2

9=

11

36

ab

(

a·cb·c

)

c 6= 0

3 + 2 = 2 + 3

3 22 35

(2 + 3) + 4 = 2 + (3 + 4)

5 + 0 = 5

(2 + 1) (1 + 2)

7 (5 + 6) 7 + (5 + 6)(7 + 5) 6 (7 + 5) + 6

$2400

(

1

2

)

(

1

6

)

(

1

4

)

12 + 1

6 + 14

2 6 4 12

2 12 :1

2=

6 · 16 · 2 =

6

12

6 12 :1

6=

2 · 12 · 6 =

2

12

4 12 :1

4=

3 · 13 · 4 =

3

126

12+

2

12+

3

12=

(6 + 2 + 3)

12=

11

12

$2, 200 $2, 400

ac

+ bc

= a+bc

ab

a·cb·c c 6= 0

2 + 3 = 3 + 2

(2 + 3) + 4 = 2 + (3 + 4)

37 + 2

7310 + 1

5516 + 5

1238 + 9

16825 + 7

1016 + 1

4715 + 2

9519 + 2

27

1

3 + 3 + 1− 8 = −1

1− 2 + 7 + 1 = 7

57

12

3548

1516

5150

512

3145

173513

112

(6) + (−6) = 0 − 6 6.

(279) + (−279) = 0 − 279 279.

(x) + (−x) = 0 − x x.

13 − 1

99 3 9

3 9 13 = 1·3

3·3 = 39

39

13

39 − 1

9

ac− b

c= a−b

c

1

3− 1

9=

2

9

2990 − 13

12690 126

90 126

90

90 = 9 · 10

9 = 3 · 310 = 5 · 290 = 3 · 3 · 5 · 2

126

126 = 9 · 14

9 = 3 · 314 = 7 · 2

126 = 3 · 3 · 7 · 2

90 126

90 126

2 1 1 13 2 2 25 1 0 17 0 1 1

2 3′ 5 7 2 · 3 · 3 · 5 · 7 = 630

90 126 630 630

90 630 7 630 90 2990 = 7·29

7·90 = 203630

126 630 5 630 126 13126 =

5·135·126 = 65

630

29

90− 13

126=

203

630− 65

630=

(203− 65)

630=

138

630

{

a

c− b

c=

a− b

c

}

138630630 138

138 = 2 · 3 · 23

138

630=

2 · 3 · 23

2 · 3 · 3 · 5 · 7 2 3

27

90− 13

126=

23

105

4

5,

5

12

19

201

6

4

5+

5

12+

19

20− 1

65, 12, 20 6.

5 = 5 5

12 = 2 · 2 · 3 2′ , 3

20 = 2 · 2 · 5 2′ , 5

6 = 2 · 3 3

2, 2, 3, 5 2 · 2 · 3 · 5 = 6060

60 5 2′ 32 · 2 · 3 = 12

4

5=

12 · 412 · 5 =

48

605

12=

5 · 55 · 12

=25

6019

20=

3 · 19

10 · 6 =57

601

6=

10 · 110 · 6 =

10

60

4860 + 25

60 + 5760 − 10

60 = (48+25+57−10)60 = 120

60

21

= 1.5t

t t = 2t = 6

( = 0 × 0.25)

0 01 1.52 33 4.54 65 7.56 9

t = 2 t = 62 3 / 6 9 /

t

t = 2= 2 (2)

(2) = 1.5(2) = 3

t = 6 6(6)

(6) = 1.5(6) = 9

t = 2 t = 6 (6)− (2) = 9− 3 = 6 /

6 /

= 3/( )2

(2) =3

(2)2=

3

4

(3) =3

(3)2=

3

9=

1

3

(3)− (2) =1

3− 3

4

3 4 12 12

1

3=

4 · 14 · 3 =

4

123

4=

3 · 33 · 4 =

9

12

4

12− 9

12=

(4− 9)

12= − 5

12

512

ac− b

c= a−b

c

512 − 9

1823 − 1

434 − 1

31511 − 9

7213 − 1

11727 − 9

39611 − 3

221364 − 7

401170 − 11

30

y = 3x + 2 y x = 3 x = 7

y = 23x + 1

2 y x = 1 x = 2

= 120speed

( )

40 90

−1125125126779

143103519229

320−22105

= +12

= −13

1 23

3 · −1 = −3

−1 · −1 = 1

x −x −x

x −x

79.5π

(x + 1)|x|

79.5 · (−1) = −79.5

π · (−1) = −π

(x + 1) · (−1) = −(x + 1) = −x− 1

|x| · (−1) = −|x|

|x| | − x| −|x|

x = −3| − 3| 6= −|3| | − 3| = 3 −|3| = −3

13 · 2

5

13 · $60

1

3· 2

5=

2

15

37 · 4

5

7 · 5 3 · 4

3

70· 4

5=

12

35

ab· c

d= ac

bd

12 · 3

425 · 5

913 · 2

7 · 25

12 · 2

3 · 34 · 4

5

1

2· 3

4=

1 · 32 · 4 =

3

8

2

5· 5

9=

2 · 55 · 9 =

2

9

1

3· 2

7· 2

5=

1 · 2 · 23 · 7 · 5 =

4

105

1

�2× �2

�3× �3

�4× �4

5=

1

5

1225 · 35

4212 42 25 35

12

25· 35

42=

6 · 225

· 35

6 · 7 =6 · 2 · 5 · 75 · 5 · 6 · 7 =

2

5

12

25· 35

42=

2

5

4 · 2 = 2 · 4

×

(2 · 3) · 4 = 2 · (3 · 4)

5 · 1 = 5

4(6 + 3) = 4 · 6 + 4 · 3

$3$5

5 ·$3 3 ·$5

$10

$10$10 · 1 $10

8× 7 3×7 5× 7

(8× 7)8 = (3 + 5) (3 + 5) · 7 (3 · 7) + (5 · 7)

(3 · 7) + (5 · 7)

15 (H2O2)H2O2

H2O2

H2O2

H2O2

H2O215 2

=1

5· 2 =

2

50.4

= 2− 2

5=

10

5− 2

5=

8

51.6

1.6

15 H2O225

= 2 · 2

5=

4

5.

=4

5+

2

5=

6

51.2

1.2 H2O2

H2O2

25 H2O2 15 2

5 H2O2

H2O2 0.15 · 25 = 0.15 · 0.4 = 0.06 H2O2

H2O2 1.2 0.061.2 = 0.05

H2O2 5

14

13

14

13

14

1.

(1)1

41

4· 1− 1

4. 1− 1

4=

4

4− 1

4=

3

41

3

(

3

4

)

1

3

1

3· 3

4=

1

4.

3

4− 1

4=

2

4=

1

21

2· 1

2=

1

4.

1

2− 1

4=

2

4− 1

4=

1

4

14

3

ab· c

d= ac

bd

2 · 3 = 3 · 2

(2 · 3) · 4 = 2 · (3 · 4)

2 · 1 = 2

4(2 + 3) = 4(2) + 4(3)

25

−105

x2

(3 + x)

(3− x)

512 × 9

10

23 × 1

4

34 × 1

3

1511 × 9

7

113 × 1

11

727 × 9

14(

35

)2

111 × 22

21 × 710

1215 × 35

13 × 102 × 26

36

−25

105

−x2

−(x + 3) −x− 3

(x− 3)

38

16

14

13577

1143

16

27125

115

709

28

= + 2 + 5

= (p + 2f + 5c)

28

= 28 · (p + 2f + 5c) 28 .

= 28(p + 2f + 5c)

28(p + 2f + 5c) = 28(p) + 28(2f) + 28(5c) = 28p + 56f + 140c

28 56 140

11(2 + 6)

11(2 + 6) = 11(8) = 88

11

11(2 + 6) = 11(2) + 11(6) = 22 + 66 = 88

11(2− 6)

11(2− 6) = 11(−4) = −44

11(2− 6) = 11(2) + 11(−6) = 22− 66 = −44

11(2x + 6)7(3x− 5)27

(

3y2 − 11)

2x7

(

3y2 − 11xy

)

11

11(2x + 6) = 22x + 66

7(3x− 5) = 21x− 35

27

(

3y2 − 11)

= 27

(

3y2)

+ 27 (−11) = 6y2

7 − 227

2

7

(

3y2 − 11)

=6y2 − 22

7

2x7

(

3y2 − 11xy

)

= 2x7 (3y2) + 2x

7

(

−11xy

)

= 6x2y

7−22�x7�xy

2x

7

(

3y2 − 11

xy

)

=6xy3 − 22

7y

2x+84

9y−23

z+62

2x+84

14 (2x + 8)

14

1

4(2x + 8) =

2x

4+

8

4=

2x

2 · 2 +4 · 24

2x + 8

4=

x

2+ 2

9y−23

13 (9y − 2)

13

1

3(9y − 2) =

9y

3− 2

3=

3 · 3x

3− 2

3

9y − 2

3= 3y − 2

3

z+62

16 (z + 6)

12

1

2(z + 6)) =

z

2+

6

2

z + 6

3=

z

2+ 3

= 8(4 + 5) .

= 8× 4 + 8× 5 = 32 + 40 .

72

$12 $7$2 $500 17

x

1 · $12

1 · $7

x · $2

= 17

= 17(12 + 7 + 2x)

17(12 + 7 + 2x) = 204 + 119 + 34x

$500

204 + 119 + 34x = 500

323 + 34x = 500 323

−323− 323

34x = 177 34

x = 5.20588 . . .

4× (6 + 3) = 4× 6 + 4× 3

(x + 4)− 2(x + 5)

12 (4z + 6)

(4 + 5)− (5 + 2)

(x + 2 + 7)

y(x + 7)

13x(3y + z)

12 (x− y)− 4

0.6(0.2x + 0.7)

6 + (x− 5) + 7

6− (x− 5) + 7

4(m + 7)− 6(4−m)

−5(y − 11) + 2y

8x+124

9x+123

11x+122

3y+26

− 6z−23

7−6p

3

65 (1 = 16 )

60

−x− 6

2z + 3

2

x + 9

xy + 7y

39xy + 13xz

x2 −

y

2 − 4

0.12x + 0.42

x + 8

18− x

10 + 4

55− 3y

2x + 3

3x + 411x2 + 6

y

2 + 13

23 − 2z

73 − 2p

35 55

13 2

4 8

2 + 0 = 2

−5 + 0 = −5

99− 0 = 99

x + 5− 5 = x

5 + (−5) = 0

2× 1 = 2

−5× 1 = −5

99÷ 1 = 99

x x

x 1x

23 × 3

2

23 × 3

2 = �2�3× �3

�2= 123

32

ab

ba

a 6= 0

37473 1

2−x

y111

37

73

49

9r

3 12 3 1

2

31

2=

3

1+

1

2=

6

2+

1

2=

7

2

312

27

−xy

− y

x111

111

111 11

111

11 x 1x

5 = 51

ab÷ c

d= a

b· d

ca

bc

d

= ab· d

c

12 ÷ 1

473 ÷ 2

3x2 ÷ 1

4y

112x÷

(

−xy

)

14

41

12 × 4

1 = 1

�2× �2.2

2 = 12

1

2÷ 1

4= 2

23

32

7

�3× �3

2 = 72

7

3÷ 2

3=

7

2

14y

4y

1x2 ×

4y

1 = x

�2× �2.2.y

1 = x.2y

1

x

2÷ 1

4y= 2xy

(

−xy

)

(

− y

x

)

112x×

(

− y

x

)

= − 11.y

2x.x

11

2x

(

−x

y

)

= −11y

2x2

=

2769.

= 69− 27 = 42

= 35 =35

60= �5.7

�5.12=

7

12

=42

(

712

) =42

1÷ 7

12

7

12

12

7.

=42

1× 12

7= �7.6

1

12

�7=

6.12

1

72

=

(

32

)

(

14

) =3

2÷ 1

4

1

4

4

1.

=3

2× 4

1=

12

2= 6

6

F = ma F m

a 713

15 .

a

a =F

m

a =

(

7 13

)

(

15

) =

(

7.3

3+

1

3

)

÷(

1

5

)

1

5

a =22

3× 5

1=

110

3

36 23 /

2

x 1x

ab

ba

ab÷ c

d= a

b× d

c

10028

− 1921

7

− z3

2xy2

52 ÷ 1

412 ÷ 7

9511 ÷ 6

712 ÷ 1

2

−x2 ÷ 5

712 ÷ x

4y(

−13

)

÷(

−35

)

72 ÷ 7

4

11÷(

−x4

)

50 18

288 38

23

27

310 /

2

= /2

1101

28

− 2119

17

− 2xy2

z3

10

914

3566

1

− 7x10

2y

x

59

2

− 44x

48 I

= 169 (1 46 40 )

= 2021

x b b2 = x

b

b2 = x (−b)·(−b) = +b2 = x,−b

x√

x 2√

x 22 = 44√

4 = ±24

(1, 4, 9, 16, 25)

121225324576121 = 11× 11

√121 = 11

225 = (5× 5)× (3× 3)

√225 = 5× 3 = 15

324 = (2× 2)× (3× 3)× (3× 3)

√324 = 2× 3× 3 = 18

576 = (2× 2)× (2× 2)× (2× 2)× (3× 3)

√576 = 2× 2× 2× 3 = 24

A√

b

848752168 = (2× 2)× 2

√8 = 2×

√2 = 2

√2

48 = (2× 2)× (2× 2)× 3

√48 = 2× 2×

√3 = 4

√3

75 = (5× 5)× 3

√75 = 5×

√3 = 5

√3

216 = (2× 2)× 2× (3× 3)× 3

√216 = 2×

√2× 3×

√3 = 6

√2√

3 = 6√

6

√a×

√b =

√ab

√a÷

√b =

√

a

b

A√

a×B√

b = AB√

ab A√

a÷B√

b =A

B

√

a

b

√8×

√2

3√

4× 4√

3√12÷

√3

12√

10÷ 6√

5√8×

√2 = 16

√8×

√2 = 4

3√

4× 4√

3 = 12√

12 = 12√

(2× 2)× 3 = 12× 2√

3

3√

4× 4√

3 = 24√

3

√12÷

√3 =

√

123 =

√4

√12÷

√3 = 2

12√

10÷ 6√

5 = 126

√

105

12√

10÷ 6√

5 = 2√

2

√2√

3√7

√ √x

√99√5√0.5√1.75

9.949874371

√8 ≈ 9.950

2.236067977

√5 ≈ 2.236

0.7071067812

√0.5 ≈ 0.707

1.322875656

√1.75 ≈ 1.323

√49 7

√49

√50√

25×√

2 = 5√

2√50

23.72.8956

π√6

3.2̄7

23.7 = 23 710

2.8956 = 2 895610000

π = 3.141592654 . . . π

√6 = 2.44949489743 . . .

√6 =

√2×

√3√

6

3.2̄7 = 3.2727272727 . . .

3.2̄7 = 3611

3.27 = 3 27100

0

−1π3√

23√369

−1π3 (π)

π3 √

23

√2

3 √369

√363 = 6

9 = 23√

369

23

−37

1735716

23 0 1

− 37 0 −1

175 2 2

5 3 4

5716 3 9

16 3 4

π227

3.14√10

10003rd 1000

π = 3.14159 . . . ≈ 3 1421000

227 = 3.14288 . . . ≈ 3 143

1000

3.14 ≈ 3 1401000√

10 = 3.16227 . . . ≈ 3 1621000

x b b2 = x b =√

x

b

√a×

√b =

√ab

√a÷

√b =

√

a

b

A√

a×B√

b = AB√

ab A√

a÷B√

b =A

B

√

a

b

√25

√24

√20

√200

√2000

√

14

√

94

√0.16

√0.1

√0.01

√13

√99

√123

√2

√2000

√0.25

√1.35

√0.37

√0.7

√0.01

√0.25

√1.35

√20

√25

√100

√6

2

61

50

√1.5

16

13

5

2√

6

2√

5

10√

2

20√

51232

0.4

1√10

√10

10

0.1

3.61

1.16

9.95

11.09

44.72

0.5

1.16

0.61

0.84

0.1

61

50

√6

2

16

13

√1.6

239594349

$18 $22.50 250$4995

$18

$22.50

$4995

250

= +

$4995 = $18 +$22.50

+ = 250

48

48

5 15 5 + 15 = 20

6 18 6 + 18 = 24

486 18

12 36 12 + 36 = 48 .

12 36

12 + 36 = 48 .

36 = 3(12) .

$80 $235$451.25

$24.50 $80 $235

$451.25

$24.50 $80 $235 $451.25

$451.25 $235 $80 $24.50

451.25− 235 + 80 + 24.50 = 320.75

$320.75

$320.75

$24.50 $320.75− $24.50 = $296.25

$80 $296.25− $80 = $216.25

$235 $216.25 + $235 = $451.25

36 16

16 4× 1636

4 16 36

36

10 4(10) + 16 = 56

9 4(9) + 16 = 52

36 16 524 9

5 4(5) + 16 = 36 .

5

5 4(5) + 16 = 36

16 4× ( ) + 1636

3616

16 4

36

16 36− 16 = 20

4 20÷ 4 = 5

5

5 4(5) + 16 = 36

$50 $0.40$70 $0.30

$50 $0.40$70 $0.30

$50 $0.40$70 $0.30

50

$50 + $0.40(50) = $70 $70 + $0.30(50) = $85

60

$50 + $0.40(60) = $74 $70 + $0.30(60) = $88

10 $15 $14

140

200

$50 + $0.40(200) = $130 $70 + $0.30(200) = $130

200

$50 + $0.40(200) = $130 $70 + $0.30(200) = $130

63.5

6

$2.25$2 17

22 $4

50

13 36

$8000 5.259 $450

16

3045

140 110

37.9 2.2

8 9

12 10

10 15

5 8

$7200 $800

48

18