Matematyka działania na potęgach i pierwiastkach

[latex] frac{ 3^{-2} }{ (1 frac{1}{4} )^{-2} } = frac{ (1 frac{1}{4}) ^{2} }{ 3^{2} } = frac{25}{9} * frac{1}{3} = frac{25}{144} [/latex]  ............................................................................. [latex] ( 4^{-3}) ^{-4} : 4^{9} = 4^{12} - 4^{9} [/latex] =[latex] 4^{3} [/latex]  ................................................................................... [latex] frac{ sqrt[3]{-16} }{3 sqrt[3]{2} } - frac{ sqrt{72} }{ sqrt{9} } = - frac{ sqrt[3]{8*2} }{3 sqrt[3]{2} } - frac{ sqrt[3]{8*9} }{ sqrt[3]{9} } = - frac{2 sqrt[3]{2} }{3 sqrt[3]{2} } - frac{2 sqrt[3]{9} }{ sqrt[3]{9} } = - frac{2}{3} -2 = -2 frac{2}{3} [/latex]  ....................................................................................... [latex] ( frac{5}{ sqrt{2} }) ^{-2} = ( frac{ sqrt{2} }{5} )^{2} = frac{2}{25} [/latex]  ........................................................................................... [latex] sqrt{ sqrt{13}+2 } * sqrt{ sqrt{13}-2 } = sqrt{( sqrt{13}+2)*( sqrt{13} -2) } = sqrt{ ( sqrt{13} )^{2} - (2)^{2} } =[/latex]  [latex] sqrt{13-4} = 3 [latex] \ \ [/latex] [latex] frac{ ( 3^{3}) ^{4} *( 3^{5} )^{-2} }{ 3^{4} } = frac{ 3^{12}* 3^{-10} }{ 3^{4} } = frac{ 3^{2} }{ 3^{4} } = 3^{-2} = frac{1}{9} [/latex]

[latex]a) \\ dfrac {3^{-2}}{(1frac14)^{-2}}=left(dfrac{3}{frac54} ight)^{-2}=(3cdotfrac45)^{-2}=(frac{12}5)^{-2}=(frac5{12})^{2}=frac{25}{144}\\\b)\\ (4^{-3})^{-4}:4^9=4^{-3cdot(-4)}:4^9= 4^{12}:4^9=4^{12-9}=4^3=64\\c)\\ dfrac{sqrt[3]{-16}}{3sqrt[3]{2}} -dfrac{sqrt[3]{72}}{sqrt[3]{9}}=dfrac{sqrt[3]{-8cdot2}}{3sqrt[3]{2}} -sqrt[3]{dfrac{72}{9}}=dfrac{-2sqrt[3]{2}}{3sqrt[3]{2}} -sqrt[3]{8}=-frac23-2=-2frac23[/latex] [latex]d)\\ left(dfrac{5}{sqrt2} ight)^{-2}=left(dfrac{sqrt2}5 ight)^2=dfrac2{25}\\\ e)\\ sqrt{sqrt{13}+2}cdotsqrt{sqrt{13}-2}=sqrt{(sqrt{13})^2-2^2}=sqrt{13-4}=sqrt9=3\\ f)\\ dfrac{(3^3)^4cdot(3^5)^{-2}}{3^4}=dfrac{3^{3cdot4}cdot3^{5cdot(-2)}}{3^4}=dfrac{3^{12+(-10)}}{3^4}=dfrac{3^{2}}{3^4}=3^{2-4}=3^{-2}=(frac13)^2=frac19[/latex]

Matematyka działania na potęgach i pierwiastkach

Matematyka działania na potęgach i pierwiastkach...