5.

x^2(y+z)^2-2xy(x-z)(y+z)+y^2(x-z)^2

=[x(y+z)-y(x-z)]^2

=(xz+yz)^2

=z^2(x+y)^2

6.

3(a+2)^2+28(a+2)-20

=[3(a+2)-2][(a+2)+ 10]

=(3a+4)(a+ 12)

7.

(a+b)^2-(b-c)^2+a^2-c^2

=(a+b)^2-c^2+a^2-(b-c)^2

=(a+b+c)(a+b-c)+(a+b-c)(a-b+c)

=(a+b-c)(a+b+c+a-b+c)

=2(a+b-c)(a+c)

8.

x(x+ 1)(x^2+x- 1)-2

=(x^2+x)(x^2+x- 1)-2

=(x^2+x)^2-(x^2+x)-2

=(x^2+x-2)(x^2+x+ 1)

=(x+2)(x- 1)(x^2+x+ 1)

9.

9x^2(x- 1)^2-3(x^2-x)-56

=9x^2(x- 1)^2-3x(x- 1)-56

=[3x(x- 1)-8][3x(x- 1)+7]

=(3x^2-3x-8)(3x^2-3x+7)

Rational number exercise

Exercise1(level b)

(1) Calculation problem:

( 1)23+(-73) (2)(-84)+(-49) (3)7+(-2.04) (4)4.23+(-7.57) (5)(-7/3)+(-7/6) (6)9/4+(-3/2) (7)3.75+(2.25)+5/4 (8)-3.75+(+5/4)+(- 1.5)

5.

x^2(y+z)^2-2xy(x-z)(y+z)+y^2(x-z)^2

=[x(y+z)-y(x-z)]^2

=(xz+yz)^2

=z^2(x+y)^2

6.

3(a+2)^2+28(a+2)-20

=[3(a+2)-2][(a+2)+ 10]

=(3a+4)(a+ 12)

7.

(a+b)^2-(b-c)^2+a^2-c^2

=(a+b)^2-c^2+a^2-(b-c)^2

=(a+b+c)(a+b-c)+(a+b-c)(a-b+c)

=(a+b-c)(a+b+c+a-b+c)

=2(a+b-c)(a+c)

8.

x(x+ 1)(x^2+x- 1)-2

=(x^2+x)(x^2+x- 1)-2

=(x^2+x)^2-(x^2+x)-2

=(x^2+x-2)(x^2+x+ 1)

=(x+2)(x- 1)(x^2+x+ 1)

9.

9x^2(x- 1)^2-3(x^2-x)-56

=9x^2(x- 1)^2-3x(x- 1)-56

=[3x(x- 1)-8][3x(x- 1)+7]

=(3x^2-3x-8)(3x^2-3x+7)

(2) Calculate by a simple method:

( 1)(- 17/4)+(- 10/3)+(+ 13/3)+( 1 1/3) (2)(- 1.8)+(+0.2)+(- 1.7)+(0. 1)+(+ 1.8)+(+ 1.4)

(3) It is known that X =+ 17 (3/4), Y =-9 (5/ 1 1) and Z =-2.25.

Find the value of: (-x)+(-y)+z.

(4) if ">" and "0" are used, then a-ba (c) if ba (d) if a.

-38)+52+ 1 18+(-62)=

(-32)+68+(-29)+(-68)=

(-2 1)+25 1+2 1+(- 15 1)=

12+35+(-23)+0=

Using rational number addition to solve the following two problems

(1) Wang Laobo took 550 yuan with him when he went to the street, went shopping in 260 yuan, and went to the bank to get 150 yuan. How much cash does Uncle Wang have on him now?

(2) The submarine originally stopped at 800 meters below the sea surface, first floated to 150 meters, and then dived to 200 meters. At this time, how many meters is the submarine under the sea?

(-6)+8+(-4)+ 12

3 and 1/4+(-2 and 3/5)+5 and 3/4+(-8 and 2/5)

9+(-7)+ 10+(-3)+(-9)

27+(-26)+33+(-27)

(+4 5/8)+(-3.257)+(-4.625)

23+(- 17)+6+(-22)

-2+3+ 1+(-3)+2+(-4)

23+(-73)

(-84)+(-49)

7+(-2.04)

4.23+(-7.57)

7/3)+(-7/6)

9/4+(-3/2)

3.75+(2.25)+5/4

-3.75+(+5/4)+(- 1.5)

(- 17/4)+(- 10/3)+(+ 13/3)+( 1 1/3)

(- 1.8)+(+0.2)+(- 1.7)+(0. 1)+(+ 1.8)+(+ 1.4)

(+ 1.3)-(+ 17/7)

(-2)-(+2/3)

|(-7.2)-(-6.3)+( 1. 1)|

|(-5/4)-(-3/4)|-| 1-5/4-|-3/4|)

(-4)(+6)(-7)

(-27)(-25)(-3)(-4)

0.00 1*(-0. 1)*( 1. 1)

24*(-5/4)*(- 12/ 15)*(-0. 12)

(-3/2)(-4/3)(-5/4)(-6/5)(-7/6)(-8/7)

(-24/7)( 1 1/8+7/3-3.75)*24

(-7 1/8)*(-23)-23*(-73/8)

(-7/ 15)*(- 18)*(-45/ 14)

(-2.2)*(+ 1.5)*(-7/ 1 1)*(-2/7)

[-|98|+76+(-87)]*23[56+(-75)-(7)]-(8+4+3)

5+2 1*8/2-6-59

68/2 1-8- 1 1*8+6 1

-2/9-7/9-56

4.6-(-3/4+ 1.6-4-3/4)

1/2+3+5/6-7/ 12

[2/3-4- 1/4*(-0.4)]/ 1/3+2

22+(-4)+(-2)+4*3

-2*8-8* 1/2+8/ 1/8

(2/3+ 1/2)/(- 1/ 12)*(- 12)

(-28)/(-6+4)+(- 1)

2/(-2)+0/7-(-8)*(-2)

( 1/4-5/6+ 1/3+2/3)/ 1/2

18-6/(-3)*(-2)

(5+3/8*8/30/(-2)-3

(-84)/2*(-3)/(-6)

1/2*(-4/ 15)/2/3

-3x+2y-5x-7y

Mixed operation of rational number addition and subtraction

Synchronous outline exercise

1. Multiple choice question:

(1) Write -2-(+3)-(-5)+(-4)+(+3) in the form of omitting brackets, and the correct one is ().

A.-2-3-5-4+3 B.-2+3+5-4+3

C.-2-3+5-4+3 D.-2-3-5+4+3

(2) The result of calculating (-5)-(+3)+(-9)-(-7)+ is correct ().

A.- 10 B.-9 C.8 D.-23

(3) The algebraic sum of-7,-12 and +2 is less than the sum of their absolute values ()

A.-38 B- 4 c . 4d . 38

(4) If +(b+3)2=0, the value of b-a- is ().

A.-4 B.-2 C.- 1 D. 1

(5) The following statement is true ()

A. the subtraction of two negative numbers is equal to the subtraction of absolute values.

B the difference between two negative numbers must be greater than zero.

C. A positive number minus a negative number is actually the algebraic sum of two positive numbers.

D. negative negative positive equals the absolute value of negative plus positive.

(6) Equation -3-5 cannot be read as ()

The difference between A.-3 and 5. The sum of b. 3 and -5.

Difference between c-3 and -5d-3 minus 5

2. Fill in the blanks: (4 ′× 4 =16 ′)

( 1)-4+7-9=- - + ;

(2)6- 1 1+4+2=- + - + ;

(3)(-5)+(+8)-(+2)-(-3)= + - + ;

(4)5-(-3 )-(+7)-2 =5+ - - + - .

3. Write the following form in the form of sum, without brackets, and say two pronunciation ways: (8 ′× 2 =16 ′)

( 1)(-2 1)+(+ 16)-(- 13)-(+7)+(-6);

(2)-2 -(- )+(-0.5)+(+2)-(+ )-2.

4. Calculation problem (6 ′× 4 = 24 ′)

( 1)- 1+2-3+4-5+6-7;

(2)-50-28+(-24)-(-22);

(3)- 19.8-(-20.3)-(+20.2)- 10.8;

(4)0.25- +(- 1 )-(+3 ).

5. When x =-3.7, y =- 1.8 and z =- 1.5, find the value of the following algebraic expression (5 ′× 4 = 20 ′).

( 1)x+y-z； (2)-x-y+z； (3)-x+y+z； (4)x-y-z。

Quality optimization training

( 1) (-7)-(+5)+(+3)-(-9)=-7 5 3 9;

(2)-(+2 )-(- 1 )-(+3 )+(- )

=( 2 )+( 1 )+( 3 )+( );

(3)- 14 5 (-3)=- 12;

(4)- 12 (-7) (-5) (-6)=- 16;

(5)b-a-(+c)+(-d)= a b c d；

2. When x=, y=- and z=- respectively, find the values of the following algebraic expressions;

( 1)x-(-y)+(-z)； (2)x+(-y)-(+z)；

(3)-(x)-y+z； (4)-x-(-y)+z。

3. Verify the equations of the following three groups of numbers:

Is it true that a-(b-c+d)=a-b+c-d?

( 1)a=-2，b=- 1，c=3，d = 5；

(2)a=23，b=-8，c=- 1，d= 1。

Computational problem

( 1)- 1-23.33-(+76.76);

(2) 1-2*2*2*2;

(3)(-6-24.3)-(- 12+9. 1)+(0-2. 1);

(4)- 1+8-7

Practical application of life

A water conservancy survey team swam 5 kilometers on the first day, 5 kilometers on the second day, 4 kilometers on the third day and 4.5 kilometers on the fourth day. Where is the survey team at this time? How many kilometers apart?

Reference answer:

Synchronous outline exercise

1.( 1)C； (2)B； (3)D； (4) a; (5)C； (6)C 2。 ( 1)4,(-7),(-9) (2)(-6),(- 1 1),(-4),2; (3)-5,8,2,3; (4)3,7,2;

3. Omit 4. ( 1)-4; (2)-80; (3)-30.5 (4)-5

5.( 1)-4; (2)4; (3)0.4; (4)-0.4.

Quality optimization training

1.( 1)-,+,+; (2)-,+,-,-; (3)+,+; (4)-,+,+; (5)-,+,-,-.

2.( 1) (2) (3) (4)-

3.( 1) and (2) both hold.

4.( 1)-

(2)

(3)-29.5

(4)- 1 Question (4) Note that the numbers with the same sign and the opposite sign should be combined separately.

Practical application of life

1. Upstream 1 km

1. 125*3+ 125*5+25*3+25

2.9999*3+ 10 1* 1 1*( 10 1-92)

3.(23/4-3/4)*(3*6+2)

4.3/7 × 49/9 - 4/3

5.8/9 × 15/36 + 1/27

6. 12× 5/6 – 2/9 ×3

7.8× 5/4 + 1/4

8.6÷ 3/8 – 3/8 ÷6

9.4/7 × 5/9 + 3/7 × 5/9

10.5/2 -( 3/2 + 4/5 )

1 1.7/8 + ( 1/8 + 1/9 )

12.9 × 5/6 + 5/6

13.3/4 × 8/9 - 1/3

14.7 × 5/49 + 3/ 14

15.6 ×( 1/2 + 2/3 )

16.8 × 4/5 + 8 × 1 1/5

17.3 1 × 5/6 – 5/6

18.9/7 - ( 2/7 – 10/2 1 )

19.5/9 × 18 – 14 × 2/7

20.4/5 × 25/ 16 + 2/3 × 3/4

2 1. 14 × 8/7 – 5/6 × 12/ 15

22. 17/32 – 3/4 × 9/24

23.3 × 2/9 + 1/3

24.5/7 × 3/25 + 3/7

25.3/ 14 ×× 2/3 + 1/6

26. 1/5 × 2/3 + 5/6

27.9/22 + 1/ 1 1 ÷ 1/2

28.5/3 × 1 1/5 + 4/3

29.45 × 2/3 + 1/3 × 15

30.7/ 19 + 12/ 19 × 5/6

3 1. 1/4 + 3/4 ÷ 2/3

32.8/7 × 2 1/ 16 + 1/2

33. 10 1 × 1/5 – 1/5 × 2 1

34.50+ 160÷40

35. 120- 144÷ 18+35

36.347+45×2-4 160÷52

37(58+37)÷(64-9×5)

38.95÷(64-45)

39. 178- 145÷5×6+42

40.8 12-700÷(9+3 1× 1 1)

4 1.85+ 14×( 14+208÷26)

43. 120-36×4÷ 18+35

44.(58+37)÷(64-9×5)

45.(6.8-6.8×0.55)÷8.5

46.0. 12× 4.8÷0. 12×4.8

47.(3.2× 1.5+2.5)÷ 1.6

48.6- 1.6÷4= 5.38+7.85-5.37=

49.7.2÷0.8- 1.2×5= 6- 1. 19×3-0.43=

50.6.5×(4.8- 1.2×4)=

5 1.5.8×(3.87-0. 13)+4.2×3.74

52.32.52-(6+9.728÷3.2)×2.5

53.[(7. 1-5.6)×0.9- 1. 15] ÷2.5

54.5.4÷[2.6×(3.7-2.9)+0.62]

55. 12×6÷( 12-7.2)-6

56. 12×6÷7.2-6

57.0.68× 1.9+0.32× 1.9

58.58+370)÷(64-45)

59.420+580-64×2 1÷28

60. 136+6×(65-345÷23)

15- 10.75×0.4-5.7

62. 18. 1+(3-0.299÷0.23)× 1

63.(6.8-6.8×0.55)÷8.5

64.0. 12× 4.8÷0. 12×4.8

65.(3.2× 1.5+2.5)÷ 1.6

66.3.2×6+( 1.5+2.5)÷ 1.6

67.0.68× 1.9+0.32× 1.9

68. 10. 15- 10.75×0.4-5.7

69.5.8×(3.87-0. 13)+4.2×3.74

70.32.52-(6+9.728÷3.2)×2.5

7 1.[(7. 1-5.6)×0.9- 1. 15] ÷2.5

72.5.4÷[2.6×(3.7-2.9)+0.62]

73. 12×6÷( 12-7.2)-6

74. 12×6÷7.2-6

75.33.02-( 148.4-90.85)÷2.5

1) 76.(25%-695%- 12%)*36

77./4*3/5+3/4*2/5

78. 1- 1/4+8/9/7/9

79.+ 1/6/3/24+2/2 1

80./ 15*3/5

8 1.3/4/9/ 10- 1/6

82./3+ 1/2)/5/6- 1/3]/ 1/7

83./5+3/5/2+3/4

84.(2-2/3/ 1/2)]*2/5

85.+5268.32-2569

86.3+456-52*8

87.5%+6325

88./2+ 1/3+ 1/4

2) 89+456-78

3) 5%+.3/7 × 49/9 - 4/3

4) 9 × 15/36 + 1/27

5) 2× 5/6 – 2/9 ×3

6) 3× 5/4 + 1/4

7) 94÷ 3/8 – 3/8 ÷6

8) 95/7 × 5/9 + 3/7 × 5/9

9) 6/2 -( 3/2 + 4/5 )

10) 8 + ( 1/8 + 1/9 )

1 1) 8 × 5/6 + 5/6

12) 1/4 × 8/9 - 1/3

13) 10 × 5/49 + 3/ 14

14) 1.5 ×( 1/2 + 2/3 )

15) 2/9 × 4/5 + 8 × 1 1/5

16) 3. 1 × 5/6 – 5/6

17) 4/7 - ( 2/7 – 10/2 1 )

18) 19 × 18 – 14 × 2/7

19) 5 × 25/ 16 + 2/3 × 3/4

20) 4 × 8/7 – 5/6 × 12/ 15

2 1) 7/32 – 3/4 × 9/24

22) 1、 2/3÷ 1/2- 1/4×2/5

2、 2-6/ 13÷9/26-2/3

3、 2/9+ 1/2÷4/5+3/8

4、 10÷5/9+ 1/6×4

5、 1/2×2/5+9/ 10÷9/20

6、 5/9×3/ 10+2/7÷2/5

7、 1/2+ 1/4×4/5- 1/8

8、 3/4×5/7×4/3- 1/2

9、 23-8/9× 1/27÷ 1/27

10、 8×5/6+2/5÷4

1 1、 1/2+3/4×5/ 12×4/5

12、 8/9×3/4-3/8÷3/4

13、 5/8÷5/4+3/23÷9/ 1 1

23) 1.2×2.5+0.8×2.5

24) 8.9× 1.25-0.9× 1.25

25) 12.5×7.4×0.8

26) 9.9×6.4-(2.5+0.24)(27) 6.5×9.5+6.5×0.5

0.35× 1.6+0.35×3.4

0.25×8.6×4

6.72-3.28- 1.72

0.45+6.37+4.55

5.4+6.9×3-(25-2.5)2×4 1846-620-380

4.8×46+4.8×54

0.8+0.8×2.5

1.25×3.6×8×2.5- 12.5×2.4

28× 12.5- 12.5×20

23.65-(3.07+3.65)

(4+0.4×0.25)8×7× 1.25

1.65×99+ 1.65

27.85-(7.85+3.4)

48× 1.25+50× 1.25×0.2×8

7.8×9.9+0.78

( 10 10+309+4+68 1+6)× 12

3×9 146×782×6×854

5. 15×7/8+6. 1-0.60625

1.3/7 × 49/9 - 4/3

2.8/9 × 15/36 + 1/27

3. 12× 5/6 – 2/9 ×3

4.8× 5/4 + 1/4

5.6÷ 3/8 – 3/8 ÷6

6.4/7 × 5/9 + 3/7 × 5/9

7.5/2 -( 3/2 + 4/5 )

8.7/8 + ( 1/8 + 1/9 )

9.9 × 5/6 + 5/6

10.3/4 × 8/9 - 1/3

1 1.7 × 5/49 + 3/ 14

12.6 ×( 1/2 + 2/3 )

13.8 × 4/5 + 8 × 1 1/5

14.3 1 × 5/6 – 5/6

15.9/7 - ( 2/7 – 10/2 1 )

16.5/9 × 18 – 14 × 2/7

17.4/5 × 25/ 16 + 2/3 × 3/4

18. 14 × 8/7 – 5/6 × 12/ 15

19. 17/32 – 3/4 × 9/24

20.3 × 2/9 + 1/3

2 1.5/7 × 3/25 + 3/7

22.3/ 14 ×× 2/3 + 1/6

23. 1/5 × 2/3 + 5/6

24.9/22 + 1/ 1 1 ÷ 1/2

25.5/3 × 1 1/5 + 4/3

26.45 × 2/3 + 1/3 × 15

27.7/ 19 + 12/ 19 × 5/6

28. 1/4 + 3/4 ÷ 2/3

29.8/7 × 2 1/ 16 + 1/2

30. 10 1 × 1/5 – 1/5 × 2 1

3 1.50+ 160÷40 (58+370)÷(64-45)

32. 120- 144÷ 18+35

33.347+45×2-4 160÷52

34(58+37)÷(64-9×5)

35.95÷(64-45)

36. 178- 145÷5×6+42 420+580-64×2 1÷28

37.8 12-700÷(9+3 1× 1 1) ( 136+64)×(65-345÷23)

38.85+ 14×( 14+208÷26)

39.(284+ 16)×(5 12-8208÷ 18)

40. 120-36×4÷ 18+35

4 1.(58+37)÷(64-9×5)

42.(6.8-6.8×0.55)÷8.5

43.0. 12× 4.8÷0. 12×4.8

44.(3.2× 1.5+2.5)÷ 1.6 (2)3.2×( 1.5+2.5)÷ 1.6

45.6- 1.6÷4= 5.38+7.85-5.37=

46.7.2÷0.8- 1.2×5= 6- 1. 19×3-0.43=

47.6.5×(4.8- 1.2×4)= 0.68× 1.9+0.32× 1.9

48. 10. 15- 10.75×0.4-5.7

49.5.8×(3.87-0. 13)+4.2×3.74

50.32.52-(6+9.728÷3.2)×2.5

5 1.[(7. 1-5.6)×0.9- 1. 15] ÷2.5

52.5.4÷[2.6×(3.7-2.9)+0.62]

53. 12×6÷( 12-7.2)-6 (4) 12×6÷7.2-6

102×4.5

7.8×6.9+2.2×6.9

5.6×0.25

8×(20- 1.25)

1) 127+352+73+44 (2)89+276+ 135+33

( 1)25+7 1+75+29 +88 (2)243+89+ 1 1 1+57

9405-2940÷28×2 1

920- 1680÷40÷7

690+47×52-398

148+3328÷64-75

360×24÷32+730

2 100-94+48×54

5 1+(2304-2042)×23

42 15+(436 1-7 16)÷8 1

(247+ 18)×27÷25

36-720÷(360÷ 18)

1080÷(63-54)×80

(528+9 12)×5-6 178

8528÷4 1×38-904

264+3 18-8280÷69

( 174+209)×26- 9000

8 14-(278+322)÷ 15

1406+735×9÷45

3 168-7828÷38+504

796-5040÷(630÷7)

285+(3000-372)÷36

1+5/6- 19/ 12

3x(-9)+7x(-9

(-54)x 1/6x(- 1/3)

1. 18. 1+(3-0.299÷0.23)× 1

2.(6.8-6.8×0.55)÷8.5

3.0. 12× 4.8÷0. 12×4.8

4.(3.2× 1.5+2.5)÷ 1.6 (2)3.2×( 1.5+2.5)÷ 1.6

5.6- 1.6÷4= 5.38+7.85-5.37=

6.7.2÷0.8- 1.2×5= 6- 1. 19×3-0.43=

7.6.5×(4.8- 1.2×4)= 0.68× 1.9+0.32× 1.9

8. 10. 15- 10.75×0.4-5.7

9.5.8×(3.87-0. 13)+4.2×3.74

10.32.52-(6+9.728÷3.2)×2.5

1 1.[(7. 1-5.6)×0.9- 1. 15] ÷2.5

12.5.4÷[2.6×(3.7-2.9)+0.62]

13. 12×6÷( 12-7.2)-6

14. 12×6÷7.2-6

15.33.02-( 148.4-90.85)÷2.5

7×(5/2 1+9/7 14)

1.8 1÷9÷9=

2.54÷6÷3=

3.38-9=

4.4 1÷5=

5.52-7=

6.77-70=

7.54+20=

8.9×3=

9.700+ 1000=

10.5080-80=

430+300

580-90

220+80

8×5-20

50-26

1000-700

63+42

35+5×6

720-650

670+300

260+ 150

63÷(86-79)

540+400

1000-560

360-5×8

45- 15÷5

100-(25+75)

839- 152+67

9×8

27+ 123-250

1000-425- 137

6 15-353- 187

8 1÷9

20÷8

45÷6

35÷7

60÷7

1 1. 1500-800=

12.4800-900=

13.6 10-30=

14.83-27=

15.80+720=

16. 1400- 1200=

17.578+76=

18.567+432=

19.90- 15=

20.45+36=

2 1.75+23=

22.35+ 17=

23.280-30=

24.8×6-9=

25.40+600=

26.4+7×9=

27.530+70=

28.(24- 16)×8=

29.(9-3)÷3=

30.4500-600=

3 1. 17+69=

32.45-7=

33.5005+ 150=

34.23 10+ 1270=

35.40+600=

36.37+26=

37.82-49=

38.5×9+7=

39.37+26=

40.8×4+28=

4 1. 100-86+34=

42.32+25+4 1=

43.26+28=

44.73+37=

45.97-88=

46.7×8+4=

36+48-25

(34-27)×5

96-6×8

29+42÷7

43+26- 17

48÷8×5

37-27÷9

(26- 19)×6

64÷(24- 16)

36+48-25

96-6×8

(34-27)×6

29+42÷7

5×( 100-99)

345-300÷9

63÷(34-25)

(83-83)÷9

43-42÷6

36÷9×8

48÷( 16÷2)

736+3287+ 1797

90 10-3875-2358

357+98-398

5×(4007-3998)

(345-396)÷7

4570-(2390+47)

280+400○280+40 3200-200○3200+200

360+90○90+360 880-90○800-90

420-300○420-30 387+595○299+399

( 19- 10)+(- 1235)

10- 10÷5×78

(36÷9-78)×4556

40-20÷4×89

6×( 18-9)÷6

4×7+ 1 ×45

(4×8+ 1)×56+(-46)÷2

825-387+659×(-568)

249+367×78+9

10 10-398

396+2 17×56

839- 152+67

1000-425- 137

327+495+ 123

6 15-353- 187

937-(37+ 16)

80 1- 187+245

72÷(300-292)

240-8×5

45÷(47-38)