Răspuns :
[tex]\displaystyle 6a).(-2)^{-4} \cdot (-2)^{-2}:2^{-3}= \frac{1}{2^4} \cdot \frac{1}{2^2} : \frac{1}{2^3} = \frac{1}{16} \cdot \frac{1}{4} : \frac{1}{8} = \frac{1}{16} \cdot \frac{1}{4} \cdot 8= \\ \\ = \frac{1}{64} \cdot 8= \frac{8}{64} = \frac{1}{8} [/tex]
[tex]\displaystyle b).6^{-3}:(-6)^{-8}:36^2= \frac{1}{6^3} : \frac{1}{6^8} :1296= \frac{1}{216} : \frac{1}{1679616} :1296= \\ \\ = \frac{1}{216} \cdot 1679616 \cdot \frac{1}{1296} = \frac{1679616}{216} \cdot \frac{1}{1296} = \frac{1679616}{279936} =6[/tex]
[tex]\displaystyle c).3^{-10} \cdot 3^2:(-27)^{-2}= \frac{1}{3^{10}} \cdot 9: \frac{1}{27^2} = \frac{1}{59049} \cdot 9: \frac{1}{729} = \\ \\ = \frac{9}{59049} \cdot 729= \frac{6561}{59049} = \frac{1}{9} [/tex]
[tex]\displaystyle d).5^{-4} \cdot \frac{1}{25} \cdot 625^{-4}: \frac{1}{5^1^2} = \frac{1}{5^4} \cdot \frac{1}{25} \cdot \frac{1}{625^4} \cdot 5^1^2= \\ \\ = \frac{1}{5^4} \cdot \frac{1}{5^2} \cdot \frac{1}{(5^4)^4} \cdot 5^1^2=\frac{1}{5^4} \cdot \frac{1}{5^2} \cdot \frac{1}{5^1^6} \cdot 5^1^2= \frac{1}{5^{4+2+16}} \cdot 5^1^2= \\ \\ = \frac{1}{5^2^2} \cdot 5^1^2= \frac{5^{12}}{5^2^2} =5^{-10}[/tex]
[tex]\displaystyle e).\left( \frac{1}{27} \right)^4 \cdot [(-3)^2]^3 \cdot 9: \left( \frac{1}{3^{-1}} \right)^6= \frac{1}{27^4} \cdot \left( \frac{1}{3^2} \right )^3\cdot 9:\left( \frac{1}{ \frac{1}{3} } \right)^6= \\ \\ = \frac{1}{27^4} \cdot \left( \frac{1}{9} \right)^3 \cdot 9 : \left( \frac{1}{3} \right)^6= \frac{1}{27^4} \cdot \frac{1}{9^3} \cdot 9: \frac{1}{3^6} = \frac{1}{531441} \cdot \frac{1}{729} \cdot 9: \frac{1}{729} = [/tex]
[tex]\displaystyle = \frac{1}{531441} \cdot \frac{1}{729} \cdot 9\cdot 729= \frac{9}{531441} = \frac{1}{59049} [/tex]
[tex]\displaystyle f).100^5 \cdot \left(- \frac{1}{10} \right)^6:[(-10)^{-1}]^{-2} \cdot \frac{1}{1000} =100^5 \cdot \frac{1}{1000000} :100 \cdot \frac{1}{1000} = \\ \\ =10000000000 \cdot \frac{1}{1000000} \cdot \frac{1}{100} \cdot \frac{1}{1000} = \frac{10000000000}{100000000} \cdot \frac{1}{1000} = \\ \\ = \frac{10000000000 }{100000000000} = \frac{1}{10}[/tex]
[tex]\displaystyle g).\left( \frac{1}{4} \right )^{-3} \cdot \frac{1}{4^8} \cdot 4^5\cdot 4^{-6}= 4^3 \cdot \frac{1}{6536}\cdot 1024 \cdot \frac{1}{4^6} = \\ \\ =64 \cdot \frac{1}{6536} \cdot 1024 \cdot \frac{1}{4096} = \frac{64}{6536} \cdot \frac{1024}{4096} = \frac{65536}{26771456} = \frac{2}{817} [/tex]
[tex]\displaystyle h). \frac{\not9^3 \cdot 3^{-2} \cdot (-3)^4 \cdot 27^{-3}}{\left( \frac{1}{3} \right )^{-2}\cdot \not9^3} = \frac{ \frac{1}{9} \cdot 81 \cdot \frac{1}{19683} }{9} = \frac{ \frac{81}{9} \cdot \frac{1}{19683} }{9} = \\ \\ = \frac{9 \cdot \frac{1}{19683} }{9} = \frac{ \frac{9}{19683} }{9}= \frac{9}{19683} :9= \frac{\not9}{19683} \cdot \frac{1}{\not 9}= \frac{1}{19683}[/tex]
[tex]\displaystyle i).(-5)^{-2} \cdot \left(- \frac{1}{5} \right)^{-3} \cdot (-5)^4 : \left( \frac{1}{5} \right)^{-5}= \frac{1}{5^2} \cdot (-5^3) \cdot 625:5^5= \\ \\ = \frac{1}{25} \cdot (-125) \cdot 625:3125=- \frac{125}{25} \cdot 625:3125=-5 \cdot 625:3125= \\ \\ =-3125:3125=-1 [/tex]
[tex]\displaystyle j).\left[\left(- \frac{1}{2}\right)^{-3}\cdot \left(2- \frac{5}{2} \right)^{-2} \right]^2: \left(6 \frac{1}{2} -7\right)^{-7}= [/tex]
[tex]\displaystyle \left[-8 \cdot \left( \frac{4-5}{2} \right)^{-2}\right]^{2}:\left( \frac{13}{2} -7\right)^{-7}=\left[-8 \cdot \left(- \frac{1}{2} \right)^{-2}\right]^2:\left( \frac{13-14}{2} \right)^{-7} \\ \\ =\left(-8 \cdot 4\right)^2:\left(- \frac{1}{2} \right)^{-7}=(-32)^2:(- 2^7)= 1024:(-128)=-8[/tex]
[tex]\displaystyle k).\left( \frac{1}{2} - \frac{2}{3} \right)^{-6} \cdot \left( \frac{2}{3} - \frac{1}{2} \right)^4: \frac{1}{36^{-1}} =\left( \frac{3}{6} - \frac{4}{6} \right)^{-6}\cdot \left( \frac{4}{6} - \frac{3}{6} \right)^4: \frac{1}{ \frac{1}{36} }= [/tex]
[tex]\displaystyle =\left(- \frac{1}{6} \right)^{-6} \cdot \left( \frac{1}{6} \right)^4: \frac{1}{36} =6^6 \cdot \frac{1}{6^4} : \frac{1}{ \frac{1}{36} } =46656 \cdot \frac{1}{1296}: 36 = \\ \\ = \frac{46656}{1296} :36 =36 : 36=1 [/tex]
[tex]\displaystyle l).\left(1 \frac{1}{2} \right)^{-4}:\left(0,25+1 \frac{1}{4} \right):\left( \frac{16}{81} \right)^{-1}=\left( \frac{3}{2} \right)^{-4}:\left( \frac{25}{100} + \frac{5}{4} \right): \frac{81}{16} = \\ \\ = \frac{16}{81} :\left( \frac{25}{100} + \frac{125}{100} \right): \frac{81}{16} = \frac{16}{81} : \frac{150}{100} : \frac{81}{16} = \frac{16}{81} \cdot \frac{100}{150} \cdot \frac{16}{81} = \\ \\ = \frac{1600}{12150} \cdot \frac{16}{81} = \frac{25600}{984150}= \frac{512}{19683}[/tex]
[tex]\displaystyle m).|-2|^{-4} \cdot \frac{1}{8^{-3}} :|-32|=2^{-4} \cdot \frac{1}{ \frac{1}{8^3} } :32= \frac{1}{2^4} \cdot \frac{1}{ \frac{1}{512} } :32= \\ \\ = \frac{1}{16} \cdot 512:32= \frac{512}{16} :32=32:32=1[/tex]
[tex]\displaystyle b).6^{-3}:(-6)^{-8}:36^2= \frac{1}{6^3} : \frac{1}{6^8} :1296= \frac{1}{216} : \frac{1}{1679616} :1296= \\ \\ = \frac{1}{216} \cdot 1679616 \cdot \frac{1}{1296} = \frac{1679616}{216} \cdot \frac{1}{1296} = \frac{1679616}{279936} =6[/tex]
[tex]\displaystyle c).3^{-10} \cdot 3^2:(-27)^{-2}= \frac{1}{3^{10}} \cdot 9: \frac{1}{27^2} = \frac{1}{59049} \cdot 9: \frac{1}{729} = \\ \\ = \frac{9}{59049} \cdot 729= \frac{6561}{59049} = \frac{1}{9} [/tex]
[tex]\displaystyle d).5^{-4} \cdot \frac{1}{25} \cdot 625^{-4}: \frac{1}{5^1^2} = \frac{1}{5^4} \cdot \frac{1}{25} \cdot \frac{1}{625^4} \cdot 5^1^2= \\ \\ = \frac{1}{5^4} \cdot \frac{1}{5^2} \cdot \frac{1}{(5^4)^4} \cdot 5^1^2=\frac{1}{5^4} \cdot \frac{1}{5^2} \cdot \frac{1}{5^1^6} \cdot 5^1^2= \frac{1}{5^{4+2+16}} \cdot 5^1^2= \\ \\ = \frac{1}{5^2^2} \cdot 5^1^2= \frac{5^{12}}{5^2^2} =5^{-10}[/tex]
[tex]\displaystyle e).\left( \frac{1}{27} \right)^4 \cdot [(-3)^2]^3 \cdot 9: \left( \frac{1}{3^{-1}} \right)^6= \frac{1}{27^4} \cdot \left( \frac{1}{3^2} \right )^3\cdot 9:\left( \frac{1}{ \frac{1}{3} } \right)^6= \\ \\ = \frac{1}{27^4} \cdot \left( \frac{1}{9} \right)^3 \cdot 9 : \left( \frac{1}{3} \right)^6= \frac{1}{27^4} \cdot \frac{1}{9^3} \cdot 9: \frac{1}{3^6} = \frac{1}{531441} \cdot \frac{1}{729} \cdot 9: \frac{1}{729} = [/tex]
[tex]\displaystyle = \frac{1}{531441} \cdot \frac{1}{729} \cdot 9\cdot 729= \frac{9}{531441} = \frac{1}{59049} [/tex]
[tex]\displaystyle f).100^5 \cdot \left(- \frac{1}{10} \right)^6:[(-10)^{-1}]^{-2} \cdot \frac{1}{1000} =100^5 \cdot \frac{1}{1000000} :100 \cdot \frac{1}{1000} = \\ \\ =10000000000 \cdot \frac{1}{1000000} \cdot \frac{1}{100} \cdot \frac{1}{1000} = \frac{10000000000}{100000000} \cdot \frac{1}{1000} = \\ \\ = \frac{10000000000 }{100000000000} = \frac{1}{10}[/tex]
[tex]\displaystyle g).\left( \frac{1}{4} \right )^{-3} \cdot \frac{1}{4^8} \cdot 4^5\cdot 4^{-6}= 4^3 \cdot \frac{1}{6536}\cdot 1024 \cdot \frac{1}{4^6} = \\ \\ =64 \cdot \frac{1}{6536} \cdot 1024 \cdot \frac{1}{4096} = \frac{64}{6536} \cdot \frac{1024}{4096} = \frac{65536}{26771456} = \frac{2}{817} [/tex]
[tex]\displaystyle h). \frac{\not9^3 \cdot 3^{-2} \cdot (-3)^4 \cdot 27^{-3}}{\left( \frac{1}{3} \right )^{-2}\cdot \not9^3} = \frac{ \frac{1}{9} \cdot 81 \cdot \frac{1}{19683} }{9} = \frac{ \frac{81}{9} \cdot \frac{1}{19683} }{9} = \\ \\ = \frac{9 \cdot \frac{1}{19683} }{9} = \frac{ \frac{9}{19683} }{9}= \frac{9}{19683} :9= \frac{\not9}{19683} \cdot \frac{1}{\not 9}= \frac{1}{19683}[/tex]
[tex]\displaystyle i).(-5)^{-2} \cdot \left(- \frac{1}{5} \right)^{-3} \cdot (-5)^4 : \left( \frac{1}{5} \right)^{-5}= \frac{1}{5^2} \cdot (-5^3) \cdot 625:5^5= \\ \\ = \frac{1}{25} \cdot (-125) \cdot 625:3125=- \frac{125}{25} \cdot 625:3125=-5 \cdot 625:3125= \\ \\ =-3125:3125=-1 [/tex]
[tex]\displaystyle j).\left[\left(- \frac{1}{2}\right)^{-3}\cdot \left(2- \frac{5}{2} \right)^{-2} \right]^2: \left(6 \frac{1}{2} -7\right)^{-7}= [/tex]
[tex]\displaystyle \left[-8 \cdot \left( \frac{4-5}{2} \right)^{-2}\right]^{2}:\left( \frac{13}{2} -7\right)^{-7}=\left[-8 \cdot \left(- \frac{1}{2} \right)^{-2}\right]^2:\left( \frac{13-14}{2} \right)^{-7} \\ \\ =\left(-8 \cdot 4\right)^2:\left(- \frac{1}{2} \right)^{-7}=(-32)^2:(- 2^7)= 1024:(-128)=-8[/tex]
[tex]\displaystyle k).\left( \frac{1}{2} - \frac{2}{3} \right)^{-6} \cdot \left( \frac{2}{3} - \frac{1}{2} \right)^4: \frac{1}{36^{-1}} =\left( \frac{3}{6} - \frac{4}{6} \right)^{-6}\cdot \left( \frac{4}{6} - \frac{3}{6} \right)^4: \frac{1}{ \frac{1}{36} }= [/tex]
[tex]\displaystyle =\left(- \frac{1}{6} \right)^{-6} \cdot \left( \frac{1}{6} \right)^4: \frac{1}{36} =6^6 \cdot \frac{1}{6^4} : \frac{1}{ \frac{1}{36} } =46656 \cdot \frac{1}{1296}: 36 = \\ \\ = \frac{46656}{1296} :36 =36 : 36=1 [/tex]
[tex]\displaystyle l).\left(1 \frac{1}{2} \right)^{-4}:\left(0,25+1 \frac{1}{4} \right):\left( \frac{16}{81} \right)^{-1}=\left( \frac{3}{2} \right)^{-4}:\left( \frac{25}{100} + \frac{5}{4} \right): \frac{81}{16} = \\ \\ = \frac{16}{81} :\left( \frac{25}{100} + \frac{125}{100} \right): \frac{81}{16} = \frac{16}{81} : \frac{150}{100} : \frac{81}{16} = \frac{16}{81} \cdot \frac{100}{150} \cdot \frac{16}{81} = \\ \\ = \frac{1600}{12150} \cdot \frac{16}{81} = \frac{25600}{984150}= \frac{512}{19683}[/tex]
[tex]\displaystyle m).|-2|^{-4} \cdot \frac{1}{8^{-3}} :|-32|=2^{-4} \cdot \frac{1}{ \frac{1}{8^3} } :32= \frac{1}{2^4} \cdot \frac{1}{ \frac{1}{512} } :32= \\ \\ = \frac{1}{16} \cdot 512:32= \frac{512}{16} :32=32:32=1[/tex]
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