Odpowiedź :
1. a) A∪B=A
A∩B =B
A\B={I,II,III,IV,V,VI,IX,X,XI,XII}
B\A=∅
b) A∪B={1,2,1/2,1/3,1/4,1/5,3,4,5,}
A∩B ={1}
A\B={2,3,4,5}
B\A={1/2,1/3,1/4,1/5}
c)A∪B={Adam,Basia,Tomek,Kasia,Ewa}
A∩B ={Tomek}
A\B={Adam,Basia}
B\A={Kasia,Ewa}
2.
(A\B)∪(C\D)={0,8,12}∪ (2,6,10}={0,2,6,8,10,12}
X\(A∪B)=x\{0,1,2,3,4,5,8,12}={1,6,7,9,10,11}
A∩(B∪C)∩D=A∩{0,1,2,3,4,5,6,8,10,12}∩D={0,2,8,12}∩D={0,8,12}
(X\A)\(C∩D)={1,,6,7,8,9,10,11,12}\{0,4,8,12}={1,6,7,9,10,11}
(C∪B)\X={0,2,4,5,6,7,8,10,11,12}\X={0}
A∩B =B
A\B={I,II,III,IV,V,VI,IX,X,XI,XII}
B\A=∅
b) A∪B={1,2,1/2,1/3,1/4,1/5,3,4,5,}
A∩B ={1}
A\B={2,3,4,5}
B\A={1/2,1/3,1/4,1/5}
c)A∪B={Adam,Basia,Tomek,Kasia,Ewa}
A∩B ={Tomek}
A\B={Adam,Basia}
B\A={Kasia,Ewa}
2.
(A\B)∪(C\D)={0,8,12}∪ (2,6,10}={0,2,6,8,10,12}
X\(A∪B)=x\{0,1,2,3,4,5,8,12}={1,6,7,9,10,11}
A∩(B∪C)∩D=A∩{0,1,2,3,4,5,6,8,10,12}∩D={0,2,8,12}∩D={0,8,12}
(X\A)\(C∩D)={1,,6,7,8,9,10,11,12}\{0,4,8,12}={1,6,7,9,10,11}
(C∪B)\X={0,2,4,5,6,7,8,10,11,12}\X={0}
Zadanie 3
a) (-122)-29-229= -122-29-229=-380
b) 726+(-227)-(-8)= 726-227+8=734-227=507
c) (-2)+(-3):( -1)-(-7)= -2-3:(-1)+7=5+7=12
d) (-2)∙3+(-4)-7∙0+1= -6-4-0+1=-9
e) (-2)∙(-2)∙ (-3)∙(-15)= 4*45=180
f) (15+ (-3)∙ (-2))∙ (14-(2+5) ∙2)=( 15+6)*(14-7*2)=21*0=0
Zadanie 4
a)= 4/3*[13/5:(-13/3)]=4/3*[13/5*(-3/13)]=4/3*[-3/5]=-4/5
b)=[ 6,6 -(2,8+3,6)]:0,2=[6,6-6,4]:0,2=0,2:0,2=1
c)= 4/5-22/5:[12*(-2)]=4/5-22/5:[-24]=4/5-22/5*(-1/24)=
4/5+11/60=48/60+11/60=59/60
d)=[(3/8-6/8):9/1000-1/3]:(-42/5)=
=[-3/8:9/1000-1/3]= [-3/8*1000/9-1/3]=[-125/3-1/3]=-126/3=-42
e) nie moge odczytać 1 mianownika
f)
Zadanie 5
a) (5³*5⁵½*5⁶¾):(5⁴ *5⁶¼)= (5⁵17/4):(5⁵17/4)=5⁵0=1
b)(2⁵5/3*2⁶(-6)):2(-6)=2⁵(-13/3):2(-6)= 2⁵⅝
c)[ (3⁵(-3)*3(-5)):(- (3⁵10)]*3⁵(6)=
- [3⁵(-8):(3⁵10]*3(⁶6)=
-[3(⁵-18)]*3(⁶6)=-3(⁶-12)
d) 3⁶⅓:(3(⁵-½)*3(⁵⅜))=3⁷⅛:3=3⁶(-2/3)
e)= 2⁶(1/8)
gdzie ⁶oznacza potęga a / - ułamki
Zadanie 6
a) [1-(2/3)³)⁶½]⁶-2= [1 -(2/3)⁶3/2]⁺-2= 1-(2/3)³=1-8/27=19/27
b)3²+(3⁸4/5*3⁸¼)= 3²+3⁵21/20=3(3+3⁵1/20)
c)=1 bo do 0 potęgi
d)[ ⅓-(3/2)⁶(-3)(2/3)³]*8=
[(⅛-8/27):8/27]*8=
[1/27*27/8]*8=
1/8*8=1
Zadanie 7
a)√(4)²-(2√3)²=√16-12=√4=2
b)³√(16)²-(2√10)²=³√256-40=³√216=6
c)√(2√7)²-(√5)²=√28-25=√3
d)(6√2₇5/2√2-2)³-[(6+5√2/2√2-2)-½(4+4√2+20]=
(6√2₇5/2√2-2)³-[(6+5√2/2√2-2)-2-2√2-10]=
Zadanie 8
a) (x-y)²/x+y , dla x=13/3, y=-1/5
(13/3+1/5)²/(13/3-1/5)=
(65/15+3/15)²/(65/15-3/15)=
(68/15)²/(62/15)=
4624/225:62/15=
4624/225*15/62=
2312/465≈4,9
b)(a+b)²:[b(a+b)]=(a+b)/b , a=-1,5 , b=1
(-1,5+1)/1=-0,5
c) x(x+2)/(x+2)²= x/(x+2) , x=√3
√3/(√3+2) = √3(√3-2)/(√3²-2²)=√3(√3-2)/-1=-√3(√3-2)
d)x(x-4)/(x+4)(x-4)= x/(x+4) , gdzie x=2
2/6=1/3
e)(x+y)²/(x-y)(x+y)=(x+y)/(x-y) , gdzie x=1/3 , y=½
(1/3+½)/(1/3-½)= (2/6+3/6)/(-1/6)=5/6:(-1/6)=-5
f)(x²-(√y)²)= (x²-y) , gdzie x=√2 ,y=2
(2-2)=0
g) (9a²-6ab+b²)-(a²-6ab+9b²)-2(a²-b²)=
9a²-6ab+b²-a²+6ab-9b²-2a²+2b²=
6a²-6b² , gdzie a=-2,b=3
6*(-2)²-6*(3)²= 24-54=-30
h)4x(x²+4xy+4y²)-4x(x²-4xy+4y²)+y²-xy=
4x³+16x²y+16xy²-4x³+16x²y-16xy²+y²-xy=
32x²y+y²-xy= y(32x²+y-x) , gdzie x=1,5 , y=-2
-2(72-2-1,5)=-2(68,5)=-137
i)[(√x)²-(√y)²]²-x²-2xy-y²= [x-y]²-x²-2xy-y²=x²-2xy+y²-x²-2xy-y²=-4xy, gdzie x=2√3, y=3
= -4*2√3*3=-24√3
j)[(2a)²-(a+3b)²]= 4a²-a²-6ab-9b²= 3a²-6ab-9b² , dla a =-1,b=1
3+6-9=0
k)x/x(x+y)=1/x+y , gdy x=1,5 , y =½
= 1/2
l)xy/y(x+y)=x/x+y=2/2-1=2
a) (-122)-29-229= -122-29-229=-380
b) 726+(-227)-(-8)= 726-227+8=734-227=507
c) (-2)+(-3):( -1)-(-7)= -2-3:(-1)+7=5+7=12
d) (-2)∙3+(-4)-7∙0+1= -6-4-0+1=-9
e) (-2)∙(-2)∙ (-3)∙(-15)= 4*45=180
f) (15+ (-3)∙ (-2))∙ (14-(2+5) ∙2)=( 15+6)*(14-7*2)=21*0=0
Zadanie 4
a)= 4/3*[13/5:(-13/3)]=4/3*[13/5*(-3/13)]=4/3*[-3/5]=-4/5
b)=[ 6,6 -(2,8+3,6)]:0,2=[6,6-6,4]:0,2=0,2:0,2=1
c)= 4/5-22/5:[12*(-2)]=4/5-22/5:[-24]=4/5-22/5*(-1/24)=
4/5+11/60=48/60+11/60=59/60
d)=[(3/8-6/8):9/1000-1/3]:(-42/5)=
=[-3/8:9/1000-1/3]= [-3/8*1000/9-1/3]=[-125/3-1/3]=-126/3=-42
e) nie moge odczytać 1 mianownika
f)
Zadanie 5
a) (5³*5⁵½*5⁶¾):(5⁴ *5⁶¼)= (5⁵17/4):(5⁵17/4)=5⁵0=1
b)(2⁵5/3*2⁶(-6)):2(-6)=2⁵(-13/3):2(-6)= 2⁵⅝
c)[ (3⁵(-3)*3(-5)):(- (3⁵10)]*3⁵(6)=
- [3⁵(-8):(3⁵10]*3(⁶6)=
-[3(⁵-18)]*3(⁶6)=-3(⁶-12)
d) 3⁶⅓:(3(⁵-½)*3(⁵⅜))=3⁷⅛:3=3⁶(-2/3)
e)= 2⁶(1/8)
gdzie ⁶oznacza potęga a / - ułamki
Zadanie 6
a) [1-(2/3)³)⁶½]⁶-2= [1 -(2/3)⁶3/2]⁺-2= 1-(2/3)³=1-8/27=19/27
b)3²+(3⁸4/5*3⁸¼)= 3²+3⁵21/20=3(3+3⁵1/20)
c)=1 bo do 0 potęgi
d)[ ⅓-(3/2)⁶(-3)(2/3)³]*8=
[(⅛-8/27):8/27]*8=
[1/27*27/8]*8=
1/8*8=1
Zadanie 7
a)√(4)²-(2√3)²=√16-12=√4=2
b)³√(16)²-(2√10)²=³√256-40=³√216=6
c)√(2√7)²-(√5)²=√28-25=√3
d)(6√2₇5/2√2-2)³-[(6+5√2/2√2-2)-½(4+4√2+20]=
(6√2₇5/2√2-2)³-[(6+5√2/2√2-2)-2-2√2-10]=
Zadanie 8
a) (x-y)²/x+y , dla x=13/3, y=-1/5
(13/3+1/5)²/(13/3-1/5)=
(65/15+3/15)²/(65/15-3/15)=
(68/15)²/(62/15)=
4624/225:62/15=
4624/225*15/62=
2312/465≈4,9
b)(a+b)²:[b(a+b)]=(a+b)/b , a=-1,5 , b=1
(-1,5+1)/1=-0,5
c) x(x+2)/(x+2)²= x/(x+2) , x=√3
√3/(√3+2) = √3(√3-2)/(√3²-2²)=√3(√3-2)/-1=-√3(√3-2)
d)x(x-4)/(x+4)(x-4)= x/(x+4) , gdzie x=2
2/6=1/3
e)(x+y)²/(x-y)(x+y)=(x+y)/(x-y) , gdzie x=1/3 , y=½
(1/3+½)/(1/3-½)= (2/6+3/6)/(-1/6)=5/6:(-1/6)=-5
f)(x²-(√y)²)= (x²-y) , gdzie x=√2 ,y=2
(2-2)=0
g) (9a²-6ab+b²)-(a²-6ab+9b²)-2(a²-b²)=
9a²-6ab+b²-a²+6ab-9b²-2a²+2b²=
6a²-6b² , gdzie a=-2,b=3
6*(-2)²-6*(3)²= 24-54=-30
h)4x(x²+4xy+4y²)-4x(x²-4xy+4y²)+y²-xy=
4x³+16x²y+16xy²-4x³+16x²y-16xy²+y²-xy=
32x²y+y²-xy= y(32x²+y-x) , gdzie x=1,5 , y=-2
-2(72-2-1,5)=-2(68,5)=-137
i)[(√x)²-(√y)²]²-x²-2xy-y²= [x-y]²-x²-2xy-y²=x²-2xy+y²-x²-2xy-y²=-4xy, gdzie x=2√3, y=3
= -4*2√3*3=-24√3
j)[(2a)²-(a+3b)²]= 4a²-a²-6ab-9b²= 3a²-6ab-9b² , dla a =-1,b=1
3+6-9=0
k)x/x(x+y)=1/x+y , gdy x=1,5 , y =½
= 1/2
l)xy/y(x+y)=x/x+y=2/2-1=2