a) [latex]x^{3}=8\x = sqrt[3]{8}\x=2[/latex] b) [latex]216x^{3}-1=0\216x^{3}=1 /:216\x^{3}=frac{1}{216}\x = sqrt[3]{frac{1}{216}}\x = frac{1}{6}[/latex] c) [latex]x^{4}=16 \x^{4}-16=0\(x^{2}-4)(x^{2}+4)=0\ (x-2)(x+2)(x^{2}+4)=0[/latex] x^2+4 ≠ 0 jest zawsze różne od zera [latex]x-2=0 vee x+2 =0\x= 2vee x=-2[/latex] d) [latex](x-3)^{3}=243\x-3 = sqrt[3]{243}\x = 3+sqrt[3]{243} = \x=3 + sqrt[3]{27cdot 9}\x = 3 + 3sqrt[3]{9}[/latex] e) [latex](x+1)^{2}=4\x^{2}+2x+1-4=0\x^{2}+2x-3 = 0\Delta =b^2-4ac = 2^2-4cdot 1cdot (-3)=4+12 = 16\sqrt{Delta }=sqrt{16}=4\x_{1}=frac{-b-sqrt{Delta }}{2a}=frac{-2-4}{2cdot 1}=-frac{6}{2}=-3\x_{2}=frac{-b+sqrt{Delta }}{2a}=frac{-2+4}{2cdot 1}=frac{2}{2}= 1[/latex] f) [latex](5x-7)^{2}=9\25x^{2}-2cdot 5xcdot7+7^{2}=9\25x^{2}-70x+49 -9= 0\25x^{2}-70x+40=0\Delta =b^2-4ac = (-70)^2-4cdot 25cdot 40=4900-4000= 900\sqrt{Delta }=sqrt{900}=30\x_{1}=frac{-b-sqrt{Delta }}{2a}=frac{70-30}{2cdot 25}=-frac{40}{50}=frac{4}{5}\x_{2}=frac{-b+sqrt{Delta }}{2a}=frac{70+30}{2cdot 25}=-frac{100}{50}=2[/latex] g) [latex]5x^{3}-525=0 /:5\x^{3}-105=0\x^{3}=105\x=sqrt[3]{105}[/latex] h) [latex]x^3-x^2=0 \x^{2}cdot (x-1)=0\x^{2}=0vee x-1=0\x=0vee x=1[/latex] i) [latex]x^{5}-8x^{2}=0\x^{2}cdot (x^{3}-8)=0\x^{2}=0vee x^{3}-8=0\x=0vee x^{3}=8\x=0vee x=sqrt[3]{8}\x=0vee x=2[/latex]
