1. Exercice 31
3
(a) 3
x 3 x = x 1/3 x 1/3 = x 2/3 = x2
(b)
2
2+ 3 = ( 2)2 + 2 2 3 + ( 3)2
=2+2 6+3=5+2 6

2. Exercice 31 (suite..)
(c)
α+ β α− β (α + β)2 + (α − β)2
+ =
α− β α+ β α2 − ( β)2
α2 + 2α β+( β)2 + α2 − 2α β+( β)2
=
α2 − β
α2 + 2α β + β + α2 − 2α β+β
=
α2 − β
2α2 + 2β
=
α2 − β
2(α2 + β)
= 2
α −β

3. Exercice 31 (suite..)
(d)
2+ 2 1+ 2
1+ +
1+ 2 1− 2
(1 + 2)(1 − 2) (2 + 2)(1 − 2) (1 + 2)(1 + 2)
= + +
(1 + 2)(1 − 2) (1 + 2)(1 − 2) (1 + 2)(1 − 2)
1−2 − 2 3+2 2
= 2−2
+ 2 + 2
1 1 −2 1 −2
−1 − 2 + 3 + 2 2
=
−1
2+ 2
=
−1
= −2 − 2

4. Exercice 31 (suite..)
(e)
x3 + x2 y + xy2 + y3 = x 2 (x + y) + y2 (x + y)
=x x+y+y x+y
= (x + y) x+y
(f)
x+y x−y x+y x−y
− = −
x−y x+y x−y x+y
x+y x+y− x−y x−y
=
x−y x+y
(x + y) − (x − y) 2y
= =
(x − y)(x + y) x 2 − y2

5. Exercice 31 (suite..)
(g)
1 x− y
=
x+ y ( x + y)( x − y)
x− y x− y
= =
( x)2 − ( y)2 x−y
(h)
h−4 h−4 h−4
=
h+4 h+4 h−4
( h − 4)2 ( h − 4)2
= =
( h)2 − 42 h − 16

6. Exercice 31 (suite..)
(i)
16x 2 − y2 16x 2 − y2 2 x + y
=
2 x− y 2 x− y2 x+ y
(16x 2 − y2 )(2 x + y)
=
4x − y
(4x − y)(4x + y)(2 x + y)
=
4x − y
= (4x + y)(2 x + y)
(j)
1 1 ∆− ∆−2
=
∆+ ∆−2 ∆+ ∆−2 ∆− ∆−2
∆− ∆−2 ∆− ∆−2
= =
∆ − (∆ − 2) 2