11 a) [latex]sqrt{50}=sqrt{25cdot2}=sqrt{25}cdotsart2=5sqrt2[/latex] b) [latex]sqrt{40}cdotsqrt{frac{2}{5}}=sqrt{40cdotfrac{2}{5}}=sqrt{16}=4[/latex] ============================= 12 a) [latex]sqrt{3}cdot sqrt{3^3}=sqrt{3cdot 3^3}= sqrt{3^4}=3^2=9[/latex] b) [latex]frac{1}{2}cdot sqrt{2}cdot sqrt{3}cdot sqrt{6}= frac{1}{2}cdot sqrt{2cdot 3cdot 6}= frac{1}{2}cdot sqrt{36}= frac{1}{2}cdot 6= 3[/latex] ============================= 14 a) [latex]5^{ frac{1}{2}}= sqrt{5}[/latex] b) [latex]7^{ frac{1}{4} }= sqrt[4]{7}[/latex] ============================= 15 a) [latex]sqrt{11}=11^{ frac{1}{2} }[/latex] b) [latex]sqrt[3]{0,5}=0,5^{ frac{1}{3} }[/latex] ============================= 18 a) [latex]x=frac{3}{4}[/latex] [latex](x^2cdot x^{frac{1}{2}}):(x^5:x^{frac{1}{2})=[/latex] [latex]x^{frac{5}{2}}:x^{frac{9}{2}}=x^{- frac{4}{2} }=x^{-2}[/latex] [latex]left(frac{3}{4} ight)^{-2}=left(frac{4}{3} ight)^{2}= frac{16}{9}=1 frac{7}{9}[/latex] b) [latex]x=frac{16}{625}[/latex] [latex]frac{x^{frac{3}{2}}cdot x^{frac{3}{4}}}{sqrt{x^5}}=frac{x^{frac{6}{4}}cdot x^{frac{3}{4}}}{x^{ frac{5}{2} }}=[/latex] [latex]frac{x^{frac{9}{4}}}{x^{ frac{5}{2} }}=frac{x^{frac{9}{4}}}{x^{ frac{10}{4} }}=x^{- frac{1}{4} }[/latex] [latex]left(frac{16}{625} ight)^{- frac{1}{4} }=left(frac{625}{16} ight)^{ frac{1}{4} }= sqrt[4]{frac{625}{16}}= frac{5}{2}=2,5[/latex] ============================= 19 a) [latex]left( frac{2}{5} ight) ^{ frac{2}{5} }cdot left( frac{5}{2} ight)^{ frac{3}{2} } =left( frac{2}{5} ight) ^{ frac{2}{5} }cdot left( frac{2}{5} ight)^{-frac{3}{2} } =[/latex] [latex]left( frac{2}{5} ight) ^{ frac{4}{10} }cdot left( frac{2}{5} ight)^{-frac{15}{10} }=left( frac{2}{5} ight)^{-frac{11}{10} }[/latex] b) [latex]frac{2 sqrt[3]{2} }{4^{ frac{1}{3} }} =frac{2cdot 2^{ frac{1}{3} } }{(2^2)^{ frac{1}{3} }} =[/latex] [latex]frac{2^{ frac{4}{3} } }{2^{ frac{2}{3} }} =2^{ frac{2}{3} }[/latex] ============================= 20 b) [latex]frac{3sqrt3cdot sqrt[4]{3} }{3^{frac{3}{4}}}=frac{3cdot 3^{ frac{1}{2} }cdot 3^{ frac{1}{4} }}{3^{frac{3}{4}}}=[/latex] [latex]frac{3cdot 3^{ frac{2}{4} }cdot 3^{ frac{1}{4} }}{3^{frac{3}{4}}}=frac{3^{ frac{7}{4} }}{3^{frac{3}{4}}}=3^1[/latex]
