[latex]frac{27^{frac{1}{3}}cdot 9^{-frac{1}{3}}}{81^{frac{1}{6}}:sqrt[3]{3}}=frac{(3^3)^{frac{1}{3}}cdot (3^2)^{-frac{1}{3}}}{(3^4)^{frac{1}{6}}:3^{frac{1}{3}}}=frac{3cdot 3^{-frac{2}{3}}}{3^{frac{2}{3}}:3^{frac{1}{3}}}=frac{3^{1+(-frac{2}{3})}}{3^{frac{2}{3}-frac{1}{3}}}=frac{3^{frac{1}{3}}}{3^{frac{1}{3}}}=3^{frac{1}{3}-frac{1}{3}} = 3^0[/latex] [latex]2log_{5}10-frac{1}{2}log_{5}16=log_{5}10^2-log_{5}16^{frac{1}{2}} = log_{5}100-log_{5}4=\=log_{5}frac{100}{4}=log_{5}25 = log_{5}5^2=2[/latex] [latex] ( 4x - 7y ) ( 4x + 7y) - 4( 2x + 1 )^2 = 16x^2-49y^2-16x^2-16x-4=\=-49y^2-16x-4[/latex] [latex]x^2ge4x\ x^2-4xge 0\ x(x-4)ge 0 \ x=0 vee x-4=0\ x=0 vee x=4\ \ x in (-infty, 0 extgreater cup extless 4, +infty)[/latex]
[latex]1.\\frac{27^{frac{1}{3}}*9^{-frac{1}{3}}}{81^{frac{1}6}:sqrt[3]{3}}=frac{(3^{3})^{frac{1}{3}}*(3^{2})^{-frac{1}{3}}}{(3^{4})^{frac{1}{6}}:3^{frac{1}{3}}}}=frac{3*3^{-frac{2}{3}}}{3^{frac{2}{3}}:3^{frac{1}{3}}}}=frac{3^{frac{1}{3}}}{3^{frac{1}{3}}}=3^{frac{1}{3}-frac{1}{3}} = 3^{0}[/latex] [latex]2.\\2log_{5}10-frac{1}{2}log_{5}16}= log_{5}10^{2}-log_{5}16^{frac{1}{2}}= log_{5}100-log_{5}4= log_{5}frac{100}{4} = \\=log_{5}25 = log_{5}5^{2} = 2[/latex] [latex]3.\\(4x-7y)(4x+7y)-4(2x+1)^{2}=16x^{2}-49y^{2}-4(4x^{2}+4x+1) =\\=16x^{2}-49y{2}-8x^{2}-16x-4 = -49y^{2}-16x-4 [/latex] 4. x² ≥ 4x x² - 4x ≥ 0 x(x - 4) ≥ 0 M. zerowe: x = 0 v x = 4 a > 0, to parabola zwrócona jest ramionami do góry x ∈ (-∞; 0 > U < 4; +∞)
