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a = √3(2√108 - √27 + √48 - 3√363) =
= √3(6√3) - 3√3 + 4√3 - 3 * 11√3) =
= √3 * √3 * (6 - 3 + 4 - 33) =
= 3 * (10 - 36) = - 3 * 26 = -78 = numar intreg
II) c)
4^(-2) : (1 / 80) + √12,25 * 5,(9) + √7 * [(1/√7) - 3√7] =
= (80 / 16) + (3,5 * 6) + (1 - 21) = 5 + 21 + 1 - 21 = 5 + 1 = 6
lll) a)
l 2√6 - 5 l = 5 - 2√6
l √13 - 2√3 l = √13 - 2√3
lll) b)
√(2√2 - 3)² + √8 = l 2√2 -3 l + 2√2 = 3 - 2√2 + 2√2 = 3
lll) c)
( l √15-4 l*(4+√15))^2005 = ((4 - √15) * (4 + √15))^2005 = (16 - 15)^2005 =1^2005 = 1
IV) a)
(√3 + 5)² = 3 + 2*5*√3 + 25 = 28 + 10√3
(3 - √2)² = 9 - 2*3*√2 + 2 = 11 - 6√2
IV) b)
(√7 + √2)² * (9 - 2√14) = (7 + 2√2√7 + 2)*(9 - 2√14) =
=
√27=3√3
√48= 4√3
3√363=11√3 *3= 33√3
a=√3 * ( 12√3 -3√3+ 4√3 -33√3)= 12*3 -3*3 +4*3 -33*3=
a=36-9+12-99= -108 +48= -60
a= -60
c)4⁻²= 1/4²= 1/16
√12,25=3,5= 3 intregi 5/10= 3 intregi 1/2= 7/2
5,(9)=5 intregi 9/9= 54/9=6
(1/√7- 3√7)= (√7/7-3√7)= (√7-21√7)/7= - 20√7/7
N= 1/16 : 1/80 + 7/2 * 6 + √7 * (-20√7/7)=
=1/16 * 80/1 + 7*3 + (-20)*7/7=
=10/2 +21 -20=
=5 +1=6
3)a) |2√6-5|= -2√6 +5= -4,89 +5= 0,11
|√13 -2√3|= -√13 +2√3= -3,60 +3,42= -0,18
b)√(2√2-3)² +√8= |2√2-3| + 2√2 = -2√2 +3 +2√2 =3
c) ( |√15-4| * (√15 +4) )²⁰⁰⁵= (15-16)²⁰⁰⁵= (-1)²⁰⁰⁵= -1
4) a) (√3 +5)²= 3+ 10√3 +25= 28+ 10√3
(3- √2)²= 9- 6√2 +2= 11-6√2
b)( √7+√2)² * (9- 2√14)= (7+2√14+2) * (9-2√14)= (9+2√14) * (9-2√14)= 81- 56 =25 ∈N
c) a=(3-2√2)(3+2√2)= 9 -8=1
b= |√5- 3| + |2 -√5|= -√5 +3 -2+√5= 3-2= 1
5)a) x/√10= x√10/10
4√8/5= 4√40/5= 4* 2√10/5= 8√10/5
x√10/10 -8√10/5 = 3 *2√5/5
x√10/10 -8√10/5= 6√5/5 aducem la acelasi numitor
x√10 -16√10= 12√5
x√10=12√5 +16√10
x= (12√5+16√10)/√10 rationalizam
x=(12√50 +16*10)/10=
x= (12*5√2+160)/10= 5(12√2 +32)/10= (12√2+32)/2= 2(6√2 +16)/2= 6√2 +16=
x=2(3√2+8)
b) A n B = [-3;√6]
A u B= [-7;4]
A - B= (√6;4]
B - A= [-7;-3)
c)( 6- 5√6)/√3= (6√3 -5√18)/3= (6√3 -5*3√2)/3= 3(2√3-5√2)/3= 2√3 -5√2
(4-2√6)/√2= (4√2- 2√12)/2= 2(2√2-√12)/2= 2√2 -2√3
(3+2√6)/√3= (3√3 +2√18)/3= (3√3+2*3√2)/3= 3(√3+2√2)/3= √3+2√2
(8+√6)/√2= (8√2+√12)/2= (8√2 + 2√3)/2= 2(4√2+√3)/2= 4√2 +√3
(2√3 -5√2 -2√2 +2√3 +√3 +2√2 -4√2-√3) * ( 4√3 +9√2)=
= ( 2√3+2√3 -5√2 -4√2) * (4√3+9√2)= (4√3 -9√2)* (4√3 -9√2)= 48 - 162= -114
6) (3x+2)/5 + (4x+3)/7 = (43x-8)/35 aducem la acelasi numitor
7(3x+2)+5(4x+3)=43x-8
21x+14 +20x+15=43x-8
41x+29=43x-8
41x-43x= -8-29
-2x= -37
2x=37
x= 37/2= 18,5