zadanie 3.1 4³*4⁴:4²=4⁷:4²=4⁵ zadanie 3.2 a) 5³*5⁻¹:5⁴=5²:5⁴=5⁻² b) 4²*8³*(0,25)²*32=(2²)²*(2³)³*(2⁻²)²*2⁵=2⁴*2⁹*2⁻⁴*2⁵=2¹⁴ c) (-3/4)⁻¹*(-1 i 1/3)⁻¹*(-2)⁻¹=(-3/4)⁻¹*(-4/3)⁻¹*(-2)⁻¹=(-4/3)¹*(-4/3)⁻¹*(-2)⁻¹= (-4/3)⁰*(-2)⁻¹=1*(-2)⁻¹=(-2)⁻¹ d) 2⁷*3⁷ 6⁷ ____ =____=6² 6³ 6⁵ zadanie 3.3 a) [a⁸:(a*a⁵)]:[a*(a⁷:a⁶)]=[a⁸:a⁶]:[a*a]=a²:a²=a⁰=1 b) [x³*(x²)³]:[(x³)⁴:x⁵]=[x³*x⁶]:[x¹²:x⁵]=x⁹:x⁷=x² zadanie 3.5 a) [(-3)²+2³]*(-5)=[9+8]*(-5)=17*(-5)= -85 b) (-6)²:(1+2³)=36:(1+8)=36:9=4 c) 12²:2⁴-(-1)⁹*0⁷*10³=144:16-(-1)*0*1000=144:16-0=9-0=9 zadanie 3.6 a) 2,5+(2/3-1,8*5/6):(-1/3)³=2,5+(2/3-9/5*5/6):(-1/27)=2,5+(2/3-3/2)*(-27)= 2,5+(-5/6)*(27)=2,5+135/6=2,5+22,5=25 b) [(6 3/8)+(1/3)⁻²*(4*3⁻¹-√9⁻¹)]⁻¹=[(6 3/8)+9*(4*1/3-1/3)]⁻¹= [(6 3/8)+9*3/3)]⁻¹=[(6 3/8)+9)]⁻¹=[51/8+9]⁻¹=[123/8]⁻¹=8/123 zadanie 3.7 a) 10⁵ 10⁰ ___ = ___ 10⁷ 10² b) 2³*2⁶ _______ = 2⁷*3²*5² 2⁹ _______ = 2⁷*3²*5² 2² _______ 3²*5² c) 2²⁰+2²⁰+2²⁰ ___________ = 4¹⁰+4¹⁰+4¹⁰ 2²⁰+2²⁰+2²⁰ _________________ = (2²)¹⁰+(2²)¹⁰+(2²)¹⁰ 2²⁰+2²⁰+2²⁰ __________ =1 2²⁰+2²⁰+2²⁰ zadanie 3.8-nie całe zadanie jest widoczne a) 4-2⁻¹*(3/5)⁰=2²-2⁻¹*1=4-1/2=3,5 b) 1/2*2⁰-3⁻²=1/2*1-3⁻²=1/3-1/9=3/9-1/9=2/9 c) [2/3+(2/3)⁻²]⁻¹=[2/3+9/4]⁻¹=[8/12+27/12]⁻¹=[35/12]⁻¹=12/35 zadanie 3.44 a) -3²*(-6)⁻¹ ________ = √1-16/25 -9*(-1/6) ________ = √9/25 3/2 ____ =3/2*5/3=5/2=2,5 3/5 b) √1 9/16 -(-2)²: 1 1/3 __________________ = -5 √25/16-4:4/3 ____________= -5 5/4-4*3/4 _________= -5 5/4-3 ______ = -5 -1,75 _____ = 0,35 -5 c) ∛27/8*√4/9 ___________ = 5⁻²*125 3/2*2/3 _______ = 1/5 5⁻²*5³ zadanie 3.45 a) √49+3√16-(√25)²=7+3*4-5=7+12-5=14 b) 6*√121-(√7)²=6*11-7=66-7=59 c) √25-9 *(-2)=√16 *(-2)=4*(-2)= -8 zadanie 3.46 a) √12:√6*√2=√2*√2=√4=2 b) √40*√5:√8=√200:√8=√25=5 c) √27:(√6:√2)=√27:√3=√9=3 d) √3:(√2*√6)=√3:√12=√3/12=√1/4=1/2 zadanie 3.47 a) √12=√4*3=2√3 b) √75=√25*3=5√3 c) ∛250=∛125*2=5∛2 d) ∛81=∛27*3=3∛3 zadanie 3.48 a) (5-√8)-(7+2√8)=5-√8-7-2√8= -2-3√8 b) (8-3√2)*√18=8√18-3√36=8√9*2-3*6=24√2-18=6(4√2-3) c) (3√8-√32):√2=(3√4*2-√16*2):√2=(6√2-4√2):√2=2√2:√2=2 zadanie 3.50 a) ∛1000-∛-343 ______________ = ∛0,001+(√0,9)² 10-(-7) _______ =17/1=17 0,1+0,9 b) ∛3 3/8-√10² _______________= (√0,5)²-∛(-0,5)³ ∛27/8-√100 __________ = 0,5-(-0,5) 3/2-10 _______ = -8,5 0,5+0,5 zadanie 3.56 a) ∛131-√36=∛131-6=∛125=5 b) √38-∛8=√38-2=√36=6 d) √270*√225*√16=√270*√225*4=√270*√900=√270*30=√8100=90 e) √110+√121=√110+11=√121=11
Własności działań na potęgach: [latex]a^{n}*a^{m}=a^{n+m}\ a^{n}:a^{m}=a^{n-m}\ (a^{n})^{m}=a^{n*m}\ a^{-n}=(frac{1}{a})^{n}\ a^{n}*b^{n}=(ab)^{n} [/latex] ==================================================================== zad 3.1 4³ * 4⁴ : 4²=4³⁺⁴⁻²=4⁵ 4⁵=(2²)⁵=2¹⁰ ====================== zad 3.2 a) 5³ * 5⁻¹ : 5⁴=5³⁻¹⁻⁴=5⁻² ------------------------------ b) 4² * 8³ * (0,25)² * 32= =(2²)² * (2³)³ * (1/4)² * 2⁵= =2⁴ * 2⁹ * (2⁻²)² * 2⁵= =2⁴ * 2⁹ * 2⁻⁴ * 2⁵= =2⁴⁺⁹⁻⁴⁺⁵= =2¹⁴ ------------------------------ c) [latex](-frac{3}{4})^{-1}*(-frac{4}{3})^{-1}*(-2)^{-1}=-frac{4}{3}*(-frac{3}{4})*(-2)^{-1}=\ =1*(-2)^{-1}=-frac{1}{2}[/latex] ------------------------------ d) [latex]frac{2^{7}*3^{7}}{6^{3}}=frac{(2*3)^{7}}{6^{3}}=frac{6^{7}}{6^{3}}=6^{4}\ frac{2^{7}*3^{7}}{6^{5}}=frac{(2*3)^{7}}{6^{5}}=frac{6^{7}}{6^{5}}=6^{2}\[/latex] ====================== zad 3.3 a) [a⁸ : (a * a⁵)]:[a * (a⁷ : a⁶)]= =[a⁸ : a¹⁺⁵]:[a * a⁷⁻⁶]= =[a⁸ : a⁶]:[a * a]= =a⁸⁻⁶ : a¹⁺¹= =a² : a²= =a²⁻²= =a⁰= =1 ------------------------------ b) [x³ * (x²)³]:[(x³)⁴ : x⁵]= =[x³ * x⁶]:[x¹² : x⁵]= =x³⁺⁶ : x¹²⁻⁵= =x⁹ : x⁷= =x⁹⁻⁷= =x² ====================== zad 3.4 Połowa liczby 2⁴⁰: 2²⁰ : 2=20²⁰⁻¹=2¹⁹ Czwatrta część liczby 4²⁰: 4²⁰ : 4=4²⁰⁻¹=4¹⁹ 4¹⁹=(2²)¹⁹=2³⁸ Odp. 2³⁸ > 2¹⁹ ====================== zad 3.5 a) [(-3)²+2³]*(-5)= =[9+8]*(-5)= =17*(-5)= =-85 ------------------------------ b) (-6)²:[1+2³]= =36:[1+8]= =36:9= =4 ------------------------------ c) 12²:2⁴-(-1)⁹ * 0⁷ * 10³= =144:16-0= =9 ====================== zad 3.6 a) [latex]2,5+(frac{2}{3}-frac{18}{10}*frac{5}{6}):(-frac{1}{3})^{3}=\ =2,5+(frac{2}{3}-frac{3}{2}):(-frac{1}{27})=\ =2,5+(frac{4}{6}-frac{9}{6})*(-27)=\ =2,5+(-frac{5}{6})*(-27)=\ =2,5+frac{45}{2}=\ =2,5+frac{225}{10}=\ =2,5+22,5=\ =25[/latex] ------------------------------ b) [latex][(6frac{3}{8}+(frac{1}{3})^{-2})*(4*3^{-1}-sqrt{9^{-1}})]^{-1}=\ =[(frac{51}{8}+9)*(4*frac{1}{3}-sqrt{frac{1}{9}})]^{-1}=\ =[(frac{51}{8}+frac{72}{8})*(frac{4}{3}-frac{1}{3})]^{-1}=\ =[frac{123}{8}*1]^{-1}=\ =frac{8}{123}[/latex] ====================== zad 3.7 a) 10⁵/10⁷=1/10²=10⁻² ------------------------------ b) (2³ * 3⁶)/(2⁷ * 3² * 5²)= =3⁴/(2⁴ * 5²) ------------------------------ c) [2²⁰+2²⁰+2²⁰]/[4¹⁰+4¹⁰+4¹⁰]= =[2²⁰+2²⁰+2²⁰]/[(2²)¹⁰+(2²)¹⁰+(2²)¹⁰]= =[2²⁰+2²⁰+2²⁰]/[2²⁰+2²⁰+2²⁰]= =1 ====================== zad 3.8 a) [latex]4-2^{-1}*(frac{3}{5})^{0}=4-frac{1}{2}*1=3frac{1}{2}[/latex] ------------------------------ b) [latex]frac{1}{2}*2^{0}-3^{-2}=frac{1}{2}*1-frac{1}{9}=frac{9}{18}-frac{2}{18}=frac{7}{18}[/latex] ------------------------------ c) [latex][frac{2}{3}+(frac{2}{3})^{-2}]^{-1}=[frac{2}{3}+frac{9}{4}]^{-1}=[frac{8}{12}+frac{27}{12}]^{-1}=(frac{35}{12})^{-1}=frac{12}{35}[/latex] ====================== zad 3.44 a) [latex]frac{-3^{2}*(-6)^{-1}}{sqrt{1-frac{16}{25}}}=frac{-9*(-frac{1}{6})}{sqrt{frac{9}{25}}}=frac{frac{3}{2}}{frac{3}{5}}=frac{3}{2}*frac{5}{3}=frac{5}{2}[/latex] ------------------------------ b) [latex]frac{sqrt{frac{25}{16}}-(-2)^{2}:frac{4}{3}}{-5}=frac{frac{5}{4}-4*frac{3}{4}}{-5}=(frac{5}{4}-frac{12}{4})*(-frac{1}{5})=-frac{7}{4}*(-frac{1}{5})=frac{7}{20}[/latex] ------------------------------ c) [latex]frac{sqrt[3]{frac{8}{27}}*sqrt{frac{4}{9}}}{5^{-2}*125}=frac{frac{2}{3}*frac{2}{3}}{frac{1}{25}*125}=frac{4}{9}*frac{1}{5}=frac{4}{45}[/latex] ====================== zad 3.45 a) √49+3√16-(√25)²= =7+3*4-25= =7+12-25= =-6 ------------------------------ b) 6*√121-(√7)²= =6*11-7= =66-7= =59 ------------------------------ c) √[25-9]*(-2)= =√16* (-2)= =4*(-2)= =-8 ====================== zad 3.46 a) √12:√6*√2=√[12:6]*√2=√2*√2=2 ------------------------------ b) √40*√5:√8=√[40*5]:√8=√200:√8=√[200:8]=√25=5 ------------------------------ c) √27:(√6:√2)=√27:(√[6:2])=√27:√3=√[27:3]=√9=3 ------------------------------ d) √3:(√2*√6)=√3:(√[2*6])=√3:√12=√3/√[4*3]=√3/2√3=1/2 ====================== zad 3.47 a) √12=√[4*3]=√4*√3=2√3 ------------------------------ b) √75=√[25*3]=√25*√3=5√3 ------------------------------ c) ∛250=∛[125*2]=∛125*∛2=5∛2 ------------------------------ d) ∛81=∛[27*3]=∛27*∛3=3∛3 ====================== zad 3.48 a) 5-√8-(7+2√8)= =5-√8-7-2√8= =-2-3√8 ------------------------------ b) (8-3√2)*√18= =8√18-3√[2*18]= =8√18-3√36= =8√18-3*6= =8√18-18 ------------------------------ c) (3√8-√32):√2= =3√[8:2]-√[32:2]= =3√4-√16= =3*2-4= =6-4= =2 ====================== zad 3.50 a) [latex]frac{sqrt[3]{1000}-sqrt[3]{-343}}{sqrt[3]{0,001}+sqrt{0,9}^{2}}=frac{10-(-7)}{0,1+0,9}=frac{10+7}{1}=17[/latex] ------------------------------ b) [latex]frac{sqrt[3]{frac{27}{8}}-sqrt{10^{2}}}{sqrt{0,5}^{2}-{sqrt[3]{(-0,5)^{3}}}}=frac{frac{3}{2}-10}{0,5-(-0,5)}=frac{1,5-10}{1}=0,5[/latex] ====================== zad a) [latex]sqrt[3]{131-sqrt{36}}=sqrt[3]{131-6}=sqrt[3]{125}=5[/latex] ------------------------------ b) [latex]sqrt{38-sqrt[3]{8}}=sqrt{38-2}=sqrt{36}=6[/latex] ------------------------------ d) [latex]sqrt{270sqrt{225sqrt{16}}}=sqrt{270sqrt{225*4}}=sqrt{270sqrt{900}}=sqrt{270*30}=\ =sqrt{8100}=90[/latex] ------------------------------ e) [latex]sqrt{110+sqrt{121}}=sqrt{110+11}=sqrt{121}=11[/latex]
