a) √40²+9²=√1600+81=√1681=41 b)√36²-28²=√1296-784=√512=16√2 c)[latex] sqrt[8]{3} * sqrt[6]{3} =3 ^{ frac{1}{8} } *3 ^{ frac{1}{6} } =3 ^{ frac{8+6}{8*6} } = 3 ^{frac{14}{48}} =3 ^{ frac{7}{24} } = sqrt[24]{3 ^{7} } [/latex] d)[latex]( frac{3}{5} ) ^{5} *( frac{5}{3} ) ^{4} =( frac{3}{5} ) ^{5} *( frac{3}{5} ) ^{-4} =frac{3}{5} [/latex] e)[latex] sqrt[5]{-32} =-2[/latex] f) [latex] sqrt[3]{-3 frac{3}{8} } = sqrt[3]{- frac{27}{8} } =- frac{3}{2} [/latex] g)[latex]125 ^{- frac{2}{3} } = frac{1}{ sqrt[3]{125^2} } = frac{1}{25} [/latex] h)√27 - √12=3√3 - 2√3=√3 i)√2 + √32 + √50 = √2 + 4√2 + 5√2 =10√2 j)(√6 - √2)² = 6 - 2√12 + 2 = 8-4√3=4(2-√3) k) (3-√5)²=9-6√5+5=14-6√5=2(7-3√5) l) (√2+√7)²=2+2√14+7=9+2√14 m) (1-√3)³=1-3√3+3*3-3√3=1-3√3+9-3√3=10-6√3=2(5-3√3)
[latex]a) sqrt{40^{2}+9^{2}} = sqrt{1600+81} = sqrt{1681} = 41[/latex] [latex]b) sqrt{36^{2}-28^{2}} = sqrt{1296-784} = sqrt{512} = sqrt{256*2}= 16sqrt{2[/latex] [latex]c) sqrt[3]3}*sqrt[6]{3} = 3^{frac{1}{3}}*3^{frac{1}{6}}=3^{frac{1}{3}+frac{1}{6}}=3^{frac{1}{2}}=sqrt{3}[/latex] [latex]d) (frac{3}{5})^{5}*(frac{5}{3})^{4} = (frac{3}{5})^{5}*(frac{3}{5})^{-4} = frac{3}{5}}[/latex] [latex]e) sqrt[5]{-32}= sqrt[5]{(-2)^{5}} = -2[/latex] [latex]f) sqrt[3]{-3frac{3}{8}} = sqrt[3]{-frac{27}{8}} = sqrt[3]{(-frac{3}{2})^{3}}= -frac{3}{2} = -1,5[/latex] [latex]g) 125^{-frac{2}{3}} = (5^{3})^{-frac{2}{3}}} = 5^{-2} = frac{1}{25}[/latex] [latex]h) sqrt{27}-sqrt{12} = sqrt{9*3}-sqrt{4*3} = 3sqrt{3}-2sqrt{3}= sqrt{3}[/latex] [latex]i) sqrt{2}+sqrt{32}+sqrt{50} = sqrt{2}+sqrt{16*2}+sqrt{25*2}=sqrt{2}+4sqrt{2}+5sqrt{2} =\\=10sqrt{2}[/latex] [latex]j) (sqrt{6}-sqrt{2})^{2} = 6-2sqrt{12}+2 = 8-2sqrt{4*3} = 8-4sqrt{3}[/latex] [latex]k) (3-sqrt5})^{2} = 9-6sqrt{5}+5 = 14-6sqrt{5}[/latex] [latex]l) (sqrt{2}+sqrt{7})^{2} = 2+2sqrt{14}+7 = 9+2sqrt{14}[/latex] [latex]m) (1-sqrt{3})^{3} = 1^{3}-3*1*sqrt{3}+3*1*3-3sqrt{3} = 1-3sqrt{3}+9-3sqrt{3}\\=10-6sqrt{3}[/latex] Zastosowane wzory: (a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² (a - b)³ = a³ - 3a²b + 3ab² - b³
