(1)Corrigé de l’exercice 1 A= −5 3

(1)

Corrigé de l’exercice 1

A= −5 3 ÷

!

−5 12 + −8

5

"

A= −5 3 ÷

#−5×5

12^×₅ +−8×12

5^×₁₂

$

A= −5 3 ÷

#−25

60 +−96 60

$

A= −5

3 ÷ −121 60 A= −5

3 × −60 121 A= −5

−1×^❍❍−3×20×^❍❍−3 121

A= 100 121

B = 7 2−10

−4 5 −10

B = 7

2 −10×2

1^×2

−4

5 − 10×5

1^×₅

B = 7 2 −20

2

−4 5 − 50

5 B = −13

2 ÷ −54 5 B = −13

2 × −5 54 B = −13

−2×^❍❍−1×5×^❍❍−1 54

B = 65 108

C = −5 3 + −5

21 ÷ −20 3

C = −5 3 + −5

21 × −3 20 C = −5

3 + −1×^✁5

−7×^❍❍−3× 1×^❍❍−3 4×^✁5

C = −5 3 + 1

28 C = −5×28

3×28

+ 1×3

28×3

C = −140 84 + 3

84

C = −137 84

A= 21×10¹⁰×200×10⁻³ 240×%

10⁸&4

A= 21×200

240 ×10¹⁰⁺⁽⁻³⁾ 10⁸^×⁴ A= 17,5×10⁷⁻³² A= 1,75×10¹×10⁻²⁵

A= 1,75×10⁻²⁴

B = 2×10⁸×210×10⁶ 12×%

10⁻⁵&2

B = 2×210

12 × 10⁸⁺⁶ 10⁻⁵^×² B = 35×10¹⁴⁻⁽⁻¹⁰⁾ B = 3,5×10¹×10²⁴

B = 3,5×10²⁵

!

124 317 = 39 843×3 + 4 788

(2)

39 843 = 4 788×8 + 1 539 4 788 = 1 539×3 + 171 1 539 = 171×9 + 0

! 124 317

39 843 124 317

39 843 = 124 317÷171 39 843÷171

= 727 233

A= (−6x−3)−(−7x+ 4) (−6x−3)

! A

A=−6x−3−(−7x+ 4) (−6x−3)

A=−6x−3−(42x²+ 21x+ (−24x) + (−12)) A=−6x−3−(42x²−3x−12)

A=−6x−3−42x²+ 3x+ 12 A=−42x²−3x+ 9

! A

A= (−6x−3)−(−7x+ 4) (−6x−3) A= (−6x−3)×1−(−7x+ 4) (−6x−3) A= (−6x−3)%

1−(−7x+ 4)&

A= (−6x−3) (1 + 7x−4) A= (−6x−3) (7x−3)

! A x= 0

A=−42x²−3x+ 9 x= 0 A=−42 ×0²−3 ×0 + 9

A= 9

A=

(3)

! A= 0

A= (−6x−3) (7x−3) (−6x−3) (7x−3) = 0

−6x−3 = 0 7x−3 = 0

−6x= 3 7x= 3 x= −3

6 x= 3

7

−1 2

3 7

! a√

b a b b

A=−4√

32−2√

8 + 5√ 18 A=−4√

16×√

2−2√ 4×√

2 + 5√ 9×√

2 A=−4×4×√

2−2×2×√

2 + 5×3×√ 2 A=−16√

2−4√

2 + 15√ 2

A=−5√ 2

B =√ 40×√

90×√ 160 B =√

4×√ 10×√

9×√ 10×√

16×√ 10 B = 2×√

10×3×√

10×4×√ 10 B = 24×

'√ 10(2

×√ 10 B = 24×10×√

10

B= 240√ 10

! a+b√

c a b c C ='

4√

5 + 2√ 6(2

C =' 4√

5(2

+ 2×4√

5×2√ 6 +'

2√ 6(2

C = 16×5 + 16√

30 + 4×6

C= 104 + 16√ 30

D=' 4√

3 + 3√ 5(2

D=' 4√

3(2

+ 2×4√

3×3√ 5 +'

3√ 5(2

D= 16×3 + 24√

15 + 9×5

D= 93 + 24√ 15

!

E ='

2 + 4√ 3( '

2−4√ 3(

E = 2²− '

4√ 3(2

E = 4−16×3 E =−44

F = 27√ 8 6√

18 F = 27×√

4×^✚✚√ 2 6×√

9×^✚✚√ 2 F = 27×2

6×3

F = 3

(4)

!

1 11

!

5 11

1 11

5 11

4 10

1 10

5 10

0 10

5 10

1 10

4 10

!

p(V,J) = 5 11 × 1

10 = 5

110 5

110

!

p(?,B) =p(B,B) +p(J,B) +p(V,B,) = 5 11 × 4

10 + 1 11 × 5

10 + 5 11 × 5

10 = 50 110

(5)

A=

−1 7 + 2 5 7+ 8 A=

−1 7 +2×7

1×7

5 7 +8×7

1×7

A=

−1 7 +14 5 7 7 +56

7 A= 13

7 ÷ 61

7 A= 13

7 × 7 61 A= 13

1×7^❆×1×7^❆ 61

A= 13 61

B = −28

9 − −28 45 ÷ 14

27 B = −28

9 − −28 45 × 27

14 B = −28

9 − −2×^✚14^✚

5×9^❆ × 3×9^❆ 1×^✚14^✚ B = −28

9 − −6 5 B = −28×5

9×5 −−6×9

5×9

B = −140

45 −−54 45

B = −86 45

C = −7 9 ÷

!

−7 5 +6

7

"

C = −7 9 ÷

#−7×7

5×7

+6×5

7×5

$

C = −7 9 ÷

#−49 35 + 30

35

$

C = −7

9 ÷ −19 35 C = −7

9 × −35 19 C = −7

−9×^❍❍−1×35×^❍❍−1 19

C = 245 171

A= 160×10³×1 500×10⁷ 0,1×%

10⁻⁸&4

A= 160×1 500

0,1 × 10³⁺⁷ 10⁻⁸^×⁴ A= 2 400 000×10¹⁰⁻⁽⁻³²⁾ A= 2,4×10⁶×10⁴²

A= 2,4×10⁴⁸

B = 36×10⁶×63×10⁻² 5 040×%

10⁷&2

B = 36×63

5 040 ×10⁶⁺⁽⁻²⁾ 10⁷^×² B = 0,45×10⁴⁻¹⁴ B = 4,5×10⁻¹×10⁻¹⁰

B = 4,5×10⁻¹¹

!

770 = 560×1 + 210 560 = 210×2 + 140

(6)

210 = 140×1 + 70 140 = 70×2 + 0

! 770

560 770

560= 770÷70 560÷70

= 11 8

A=−60x+ 100x²+ 9−(−10x+ 3) (9x+ 6)

! A

A= 100x²+ 9−60x−(−10x+ 3) (9x+ 6)

A= 100x²−60x+ 9−(−90x²+ (−60x) + 27x+ 18) A= 100x²−60x+ 9 + 90x²+ 33x−18

A= 190x²−27x−9

! A

A= 9−60x+ 100x²−(−10x+ 3) (9x+ 6) A= 100x²−60x+ 9−(−10x+ 3) (9x+ 6) A= (−10x+ 3)²−(−10x+ 3) (9x+ 6) A= (−10x+ 3)%

−10x+ 3−(9x+ 6)&

A= (−10x+ 3) (−10x+ 3−9x−6) A= (−10x+ 3) (−19x−3)

! A x= −3 2

A= 190x²−27x−9 x= −3 2 A= 190 ×

!

−3 2

"2

−27 ×

!

−3 2

"

−9

A= 95×2^✁

1 ×

9

2×^✁2+ −27

−1×^❍❍−1×

3×^❍❍−1 2 −9 A= 855

2 +81

2 + −18 2

A= 918 2 = 459

(7)

! A= 0

A= (−10x+ 3) (−19x−3) (−10x+ 3) (−19x−3) = 0

−10x+ 3 = 0 −19x−3 = 0

−10x=−3 −19x= 3 x= 3

10 x= −3

19

3 10

−3 19

! a√

b a b b

A=−3√

20−2√ 80−√

45 A=−3√

4×√

5−2√ 16×√

5−√ 9×√

5 A=−3×2×√

5−2×4×√

5−1×3×√ 5 A=−6√

5−8√

5−3√ 5 A=−17√

5

B =√ 48×√

27×√ 12 B =√

16×√ 3×√

9×√ 3×√

4×√ 3 B = 4×√

3×3×√

3×2×√ 3 B = 24×

'√ 3(2

×√ 3 B = 24×3×√

3 B= 72√

3

! a+b√

c a b c C ='

3√

7−4√ 6(2

C =' 3√

7(2

−2×3√

7×4√ 6 +'

4√ 6(2

C = 9×7−24√

42 + 16×6 C= 159−24√

42

D=' 3√

7 +√ 6(2

D=' 3√

7(2

+ 2×3√ 7×√

6 +√ 6² D= 9×7 + 6√

42 + 1×6 D= 69 + 6√

42

!

E ='

3−4√ 3( '

3 + 4√ 3( E = 3²−

' 4√

3(2

E = 9−16×3 E =−39

F = 24√ 90 9√

160 F = 24×√

9×^✟✟√^✟ 10 9×√

16×^✟✟√^✟ 10 F = 24×3

9×4 F = 2

(8)

!

4 12

!

3 12

4 12

5 12

2 11

4 11

5 11

3 11

5 11

3 11

4 11

!

p(M,V) = 5 12 × 4

11 = 20

132 20

132

!

p(?,J) =p(J,J) +p(V,J) +p(M,J,) = 3 12 × 2

11+ 4 12× 3

11+ 5 12× 3

11 = 33 132

(9)

A= (6x+ 9)²

A= (6x)²+ 2×6x×9 + 9² A= 36x²+ 108x+ 81 B = (7x−9)²

B = (7x)²−2×7x×9 + 9² B = 49x²−126x+ 81

C = (10x+ 1) (10x−1) C = (10x)²−1²

C = 100x²−1

D= (−x−3) (−6x−5) D= 6x²+ 5x+ 18x+ 15

D= 6x²+ 23x+ 15

E =−(3x+ 2) (3x−2) + (9x−10)²

E =−((3x)²−2²) + (9x)²−2×9x×10 + 10² E =−(9x²−4) + 81x²−180x+ 100

E =−9x²+ 4 + 81x²−180x+ 100 E = 72x²−180x+ 104

F = (x+ 9)²−(8x+ 3) (−9x−6)

F =x²+ 2×x×9 + 9²−(−72x²+ (−48x) + (−27x) + (−18)) F =x²+ 18x+ 81−(−72x²−75x−18)

F =x²+ 18x+ 81 + 72x²+ 75x+ 18 F = 73x²+ 93x+ 99

A= (−2x+ 4) (9x−9) + (9x−9)² A= (9x−9) (−2x+ 4 + 9x−9)

A= (9x−9) (7x−5) B = (−10x+ 10)²−100 B = (−10x+ 10)²−10²

B = (−10x+ 10 + 10) (−10x+ 10−10) B =

B = (−10x+ 20)×(−10x)

C = (−2x−4) (7x−9)−(7x−9) C = (−2x−4) (7x−9)−(7x−9)×1 C = (7x−9)!

−2x−4−1"

C = (7x−9) (−2x−5)

D=−(x+ 6) (−8x−6) + (7x+ 9) (−8x−6) D= (−8x−6)!

−(x+ 6) + 7x+ 9"

D= (−8x−6) (−x−6 + 7x+ 9) D= (−8x−6) (6x+ 3)

(10)

E = 16x²−81 E = (4x)²−9²

E = (4x+ 9) (4x−9)

F = (−2x+ 8) (4x−8) + 16x²−64

F = (−2x+ 8) (4x−8) + (4x)²−8²

F = (−2x+ 8) (4x−8) + (4x−8) (4x+ 8) F = (4x−8) (−2x+ 8 + 4x+ 8)

F = (4x−8) (2x+ 16)

A= 81x²+ 64 + 144x−(9x+ 8) (2x+ 4)

! A

A= 64 + 144x+ 81x²−(9x+ 8) (2x+ 4)

A= 81x²+ 144x+ 64−(18x²+ 36x+ 16x+ 32) A= 81x²+ 144x+ 64−18x²−52x−32

A= 63x²+ 92x+ 32

! A

A= 64 + 144x+ 81x²−(9x+ 8) (2x+ 4) A= 81x²+ 144x+ 64−(9x+ 8) (2x+ 4) A= (9x+ 8)²−(9x+ 8) (2x+ 4)

A= (9x+ 8)!

9x+ 8−(2x+ 4)"

A= (9x+ 8) (9x+ 8−2x−4) A= (9x+ 8) (7x+ 4)

! A x=−2

A= 63x²+ 92x+ 32 x=−2 A= 63 ×(−2)²+ 92 ×(−2) + 32

A= 252−184 + 32 A= 100

A=

! A= 0

A= (9x+ 8) (7x+ 4) (9x+ 8) (7x+ 4) = 0

9x+ 8 = 0 7x+ 4 = 0 9x=−8 7x=−4 x= −8

9 x= −4

7

−8 9

−4 7

(11)

−2x+ 7

4 −

4x−3

8 = 8x+ 7 6 (−2x+ 7)×6

4×6 −

(4x−3)×3

8×3

= (8x+ 7)×4

6×4

−12x+ 42−(12x−9)

✚24✚ = 32x+ 28

✚24✚

−12x+ 42−12x+ 9 = 32x+ 28

−24x+ 51 = 32x+ 28

−24x−32x= 28−51

−56x=−23 x= 23

56= 23 56

23 56

# −6x − 5y = 61 (×3)

9x − 6y = −24 (×2)

# −18x − 15y = 183 18x − 12y = −48

✘✘✘

−18x−15y^✘✘+18^✘x−12y = 183−48

−27y= 135 y= 135

−27 =−5

−6x−5y= 61 y=−5

−6x−5×(−5) = 61

−6x= 61−25

x= 36

−6 =−6

(x; y) = (−6; −5)

# −6 ×(−6)−5 ×(−5) = 36 + 25 = 61 9 ×(−6)−6 ×(−5) =−54 + 30 =−24

(12)

A= (2x+ 6)²

A= (2x)²+ 2×2x×6 + 6² A= 4x²+ 24x+ 36 B = (6x+ 1) (−2x−5)

B =−12x²+ (−30x) + (−2x) + (−5) B =−12x²−32x−5

C = (4x−5)²

C = (4x)²−2×4x×5 + 5² C = 16x²−40x+ 25 D= (2x+ 5) (2x−5) D= (2x)²−5²

D= 4x²−25

E = (3x+ 8) (3x−8) + (x−7)² E = (3x)²−8²+x²−2×x×7 + 7² E = 9x²−64 +x²−14x+ 49

E = 10x²−14x−15

F = (−10x−9) (−4x+ 10)−(8x+ 8)²

F = 40x²+ (−100x) + 36x+ (−90)−((8x)²+ 2×8x×8 + 8²) F = 40x²−64x−90−(64x²+ 128x+ 64)

F = 40x²−64x−90−64x²−128x−64 F =−24x²−192x−154

A=−(7x+ 3) (−9x+ 3) + 49x²−9 A=−(7x+ 3) (−9x+ 3) + (7x)²−3² A=−(7x+ 3) (−9x+ 3) + (7x+ 3) (7x−3) A= (7x+ 3)!

−(−9x+ 3) + 7x−3"

A= (7x+ 3) (9x−3 + 7x−3) A= (7x+ 3) (16x−6) B = 9x²−16

B = (3x)²−4²

B = (3x+ 4) (3x−4)

C = 25−(3x+ 6)² C = 5²−(3x+ 6)² C = (5 + 3x+ 6)!

5−(3x+ 6)"

C = (5 + 3x+ 6) (5−3x−6) C = (3x+ 11) (−3x−1)

D=−(−3x+ 7) (−9x−4) + (10x−6) (−3x+ 7) D= (−3x+ 7)!

−(−9x−4) + 10x−6"

D= (−3x+ 7) (9x+ 4 + 10x−6) D= (−3x+ 7) (19x−2)

E =−(4x−4) (4x+ 8) + (4x−4)²

(13)

E = (4x−4)!

−(4x+ 8) + 4x−4"

E = (4x−4) (−4x−8 + 4x−4) E = (4x−4)×(−12)

F =−(5x+ 10) (−3x−4) + (5x+ 10)

F =−(5x+ 10) (−3x−4) + (5x+ 10)×1 F = (5x+ 10)!

−(−3x−4) + 1"

F = (5x+ 10) (3x+ 4 + 1) F = (5x+ 10) (3x+ 5)

A= (5x−1)²−(−8x+ 4) (5x−1)

! A

A= (5x−1)²−(−8x+ 4) (5x−1)

A= (5x)²−2×5x×1 + 1²−(−40x²+ 8x+ 20x+ (−4)) A= 25x²−10x+ 1−(−40x²+ 28x−4)

A= 25x²−10x+ 1 + 40x²−28x+ 4 A= 65x²−38x+ 5

! A

A= (5x−1)²−(−8x+ 4) (5x−1) A= (5x−1)!

5x−1−(−8x+ 4)"

A= (5x−1) (5x−1 + 8x−4) A= (5x−1) (13x−5)

! A x= −4 7

A= 65x²−38x+ 5 x= −4 7 A= 65 ×

#

−4 7

$²

−38 ×

#

−4 7

$ + 5

A= 1 040

49 + −38

−1×^❍❍−1×4×^❍❍−1 7 + 5 A= 1 040

49 +1 064 49 +245

49

A= 2 349 49

! A= 0

A= (5x−1) (13x−5) (5x−1) (13x−5) = 0

5x−1 = 0 13x−5 = 0

(14)

5x= 1 13x= 5 x= 1

5 x= 5

13

1 5

5 13

−10x+ 2

3 + −x+ 5

6 = 5x−5 2 (−10x+ 2)×2

3×2

+ −x+ 5

6 = (5x−5)×3

2×3

−20x+ 4−x+ 5

✁6 = 15x−15 6✁

−21x+ 9 = 15x−15

−21x−15x=−15−9

−36x=−24 x= 24

36= 2 3

2 3

% −8x + 6y = 108 (×3)

−6x + 8y = 102 (×(−4))

% −24x + 18y = 324 24x − 32y = −408

✘✘✘−24x+ 18y^✘✘✘+24x−32y = 324−408

−14y=−84 y= −84

−14 = 6

−8x+ 6y= 108 y= 6

−8x+ 6×6 = 108

−8x= 108−36

x= 72

−8 =−9

(x; y) = (−9; 6)

% −8 ×(−9) + 6 ×6 = 72 + 36 = 108

−6 ×(−9) + 8 ×6 = 54 + 48 = 102