Odpowiedź :
Zad. 1
[tex]\frac13*\frac13*\frac13*\frac13*\frac13=(\frac13)^5\\Odp. A[/tex]
Zad. 2
[tex]a)\\\frac{5^2}5=5^2:5^1=5^1=5\\\\b) \\\frac{(-3)^5}3=\frac{-3^5}3=-3^4=-81\\\\c)\\\frac{4}{(-4)^2}=\frac{4}{4^2}=\frac{1}{4^1}=\frac14\\\\d)\\\frac{-2^4}{8}=\frac{-2^4}{2^3}=-2^1=-2[/tex]
Zad. 3
[tex]\frac{5^{13}:(5^6*5^2)}{5^3}=\frac{5^{13}:5^8}{5^3}=5^{13-8-3}=5^{13-11}=5^2=25[/tex]
Zad. 4
[tex]a)\\(4^3)^5*(4^2)^4=4^{15}*4^8=4^{23}\\\\b) \\(6^7)^3:(6^4)^4=6^{21}:6^{16}=6^{5}\\\\c) \\((\frac12)^5)^5*((\frac12)^4)^4=(\frac12)^{25}*(\frac12)^{16}=(\frac12)^{41}=2^{-41}\\\\d) \\(0.9^4)^9:(0.9^5)^6=0.9^{36}:0.9^{30}=0.9^6[/tex]
Zad. 5
[tex]a) \\7^9*7^8=7^{17}\\\\b) \\5^8*5*5^{10}=5^{19}\\\\c) \\(1\frac37)^9:(1\frac37)^7=(1\frac37)^2=(\frac{10}7)^2[/tex]
Zad. 6
[tex]a)\\4^8=(2^2)^8=2^{16}\\\\b) \\32^{11}=(2^5)^{11}=2^{55}\\\\c)\\(8^2)^4=8^8=(2^3)^8=2^{24}\\\\d) \\(16^4)^3=16^{12}=(2^4)^{12}=2^{48}[/tex]
Zad. 7
[tex]a)\\\\\sqrt[3]{125}+\sqrt{81}=5+9=14\\\\b) \\\\\sqrt[3]{6^2+28}=\sqrt[3]{36+28}=\sqrt[3]{64}=4\\\\c)\\\\\sqrt[3]{9^3}+\sqrt{15^2}=9+15=24\\\\d)\\\\\sqrt{12}*\sqrt3-\sqrt[3]2*\sqrt[3]4=\sqrt{36}-\sqrt[3]{8}=6-2=4\\\\e) \\\\\frac{\sqrt{72}+2\sqrt2}{\sqrt2}=\frac{\sqrt2(\sqrt{36}+2)}{\sqrt2}=6+2=8[/tex]
Zad. 8
[tex]a)\\7\sqrt6-4-3\sqrt6+7=4\sqrt6+3\\\\b) \\4(2+3\sqrt2)-3(3+2\sqrt2)=8+12\sqrt2-(9+6\sqrt2)=8+12\sqrt2-9-6\sqrt2=6\sqrt2-1\\\\c)\\\\6(\sqrt[3]5-2)-4(\sqrt[3]5-1)=6\sqrt[3]5-12-(4\sqrt[3]5-4)=6\sqrt[3]5-12-4\sqrt[3]5+4=2\sqrt[3]5-8[/tex]
Zad. 9
[tex]a)\\\\3(\sqrt5)^2=3*5=15\\\\b) \\\\3\sqrt2*5\sqrt2=3*5*2=15*2=30\\\\c) \\\\(\frac35\sqrt{10})^2=\frac9{25}*10=\frac9{5}*2=\frac{18}5=3\frac35=3,6[/tex]
Zad. 10
[tex]\frac{2^7}{5^2}*\frac{625}{32}=\frac{2^7}{5^2}*\frac{5^4}{2^5}=\frac{2^2}{1}*\frac{5^2}{1}=4*25=100[/tex]
Odpowiedź:
1.A
2.a)
[tex] \frac{ {5}^{2} }{5} = 5[/tex]
b)
[tex] \frac{( - 3)^{5} }{3} = \frac{ - 243}{3} = - 81[/tex]
c)
[tex]4 \div ( - 4) ^{2} = 4 \div 16 = \frac{1}{4} [/tex]
d)
[tex]( { - 2})^{4} \div 8 = - 16 \div 8 = - 2[/tex]
3.
[tex](( {5}^{12} \div ( {5}^{6} \times {5}^{2} )) \div {5}^{3} = ( {5}^{12} \div {5}^{8} ) \div {5}^{3} = {5}^{4} \div {5}^{3} = 5[/tex]
4.a)
[tex]( { {4}^{3} })^{5} \times ({ {4}^{2} })^{4} = {4}^{15} \times {4}^{8} = {4}^{23} [/tex]
b)
[tex]( {6}^{7} )^{3} \div ( { {6}^{4} })^{4} = {6}^{21} \div {6}^{16} = {6}^{5} [/tex]
c)
[tex](( \frac{1}{2}) ^{5} )^{5} \times (( \frac{1}{2} ) ^{4} ) ^{4} = ({ \frac{1}{2} })^{25} \times ({ \frac{1}{2} })^{16} = ({ \frac{1}{2} })^{41} [/tex]
d)
[tex]( {0.9}^{4} ) ^{9} \div ( {0.9}^{5}) ^{6} = {0.9}^{36} \div {0.9}^{30} = {0.9}^{6} [/tex]
5.a)
[tex] {7}^{9} \times {7}^{8} = {7}^{17} [/tex]
b)
[tex] {5}^{8} \times 5 \times {5}^{10} = {5}^{19} [/tex]
c)
[tex] ({1 \frac{3}{7} })^{9} \div ( {1 \frac{3}{7} })^{7} = ({1 \frac{3}{7} })^{2} = ( { \frac{10}{7} })^{2} [/tex]
6.a)
[tex] {4}^{8} = ( {2}^{2}) ^{8} = {2}^{16} [/tex]
b)
[tex] {32}^{11} = ( {2}^{5} ) ^{11} = {2}^{55} [/tex]
c)
[tex]( {8}^{2} ) ^{4} = (( {2}^{3} ) ^{2}) ^{4} = {2}^{24} [/tex]
d)
[tex]( {16}^{4} ) ^{3} = (( {2}^{4} ) ^{4}) ^{3} = {2}^{48} [/tex]
7.a)
[tex] \sqrt[3]{125} + \sqrt{81} = 5 + 9 = 14[/tex]
b)
[tex] \sqrt[3]{ {6}^{2} + 28} = \sqrt[3]{64} = 4[/tex]
c)
[tex] \sqrt[3]{ {9}^{3} } + \sqrt{ {15}^{2} } = 9 + 15 = 24[/tex]
d)
[tex] \sqrt{12} \times \sqrt{3} - \sqrt[3]{2} \times \sqrt[3]{4} = \sqrt{36} - \sqrt[3]{8} = 6 - 2 = 4[/tex]
e)
[tex]( \sqrt{72} + 2 \sqrt{2} ) \div \sqrt{2} = (6 \sqrt{2} + 2 \sqrt{2} ) \div \sqrt{2} = 8 \sqrt{2} \div \sqrt{2} = 8[/tex]
8.a)
[tex]7 \sqrt{6} - 4 - 3 \sqrt{6} + 7 = 4 \sqrt{6} + 3[/tex]
b)
[tex]4(2 + 3 \sqrt{2)} - 3(3 + 2 \sqrt{2)} = 8 + 12 \sqrt{2} - 9 + 6 \sqrt{2} = - 1 + 18 \sqrt{2} [/tex]
c)
[tex]6( \sqrt[3]{5} - 2) - 4( \sqrt[3]{5} - 1) = 6 \sqrt[3]{5} - 12 - 4 \sqrt[3]{5} + 4 = 2 \sqrt[3]{5} - 8[/tex]
9.a)
[tex]3( \sqrt{ {5}})^{2} = 3 \times 5 = 15[/tex]
b)
[tex]3 \sqrt{2} \times 5 \sqrt{2} = 15 \times 2 = 30[/tex]
c)
[tex]({ \frac{3}{5} \sqrt{10} }) ^{2} = \frac{9}{25} \times 10 = \frac{18}{5} = 3 \frac{3}{5} [/tex]
10.
[tex] \frac{ {2}^{7} }{ {5}^{2} } \times \frac{625}{32} = \frac{ {2}^{7} }{ {5}^{2} } \times \frac{ {5}^{4} }{ {2}^{5} } = {2}^{2} \times {5}^{2} = {10}^{2} = 100[/tex]