[latex]left(frac{25}{81} ight )^{x^3-frac{1}{2}}=1,8^{2x-x^2} \left(left(frac{5}{9} ight)^2 ight )^{x^3-frac{1}{2}}=left(frac{9}{5} ight )^{2x-x^2} \left(frac{5}{9} ight ) ight)^{2x^3-1}=left(frac{5}{9} ight )^{x^2-2x} \2x^3-1=x^2-2x \2x^3-x^2+2x-1=0 \x^2(2x-1)+(2x-1)=0 \(x^2+1)(2x-1)=0 \x^2+1=0 quad vee quad 2x-1=0 \x in varnothing quad vee quad x=frac{1}{2} \\oxed{x=frac{1}{2}}[/latex] [latex]left(2sqrt{2} ight )^{x+1}=left(frac{1}{4} ight )^{-x+2} \left(2^frac{3}{2} ight )^{x+1}=left(2^{-2} ight )^{-x+2} \left(2^frac{3}{2} ight )^{x+1}=left(2^{frac{3}{2}} ight )^{-frac{4}{3}(-x+2)} \x+1=-frac{4}{3}left(-x+2 ight )quad /cdot (-3) \-3x-3=-4x+8 \\oxed{x=11}[/latex] [latex]left(frac{9}{sqrt{3}} ight )^x=3^{x+1} \left(3^{frac{3}{2}} ight )^x=3^{x+1} \3^{frac{3}{2}x}=3^{x+1} \frac{3}{2}x=x+1quad /cdot 2 \3x=2x+2 \\oxed{x=2}[/latex] [latex]left(frac{81sqrt[3]{3}}{sqrt{27}} ight )^{6x}=3^{x^2} \left(frac{3^4cdot 3^{frac{1}{3}}}{3^{frac{3}{2}}} ight )^{6x}=3^{x^2} \left(3^{frac{24+2-9}{6}} ight )^{6x}=3^{x^2} \3^{17x}=3^{x^2} \17x=x^2 \x^2-17x=0 \x(x-17)=0 \x=0 quad vee quad x-17=0 \\oxed{x=0 quad vee quad x=17}[/latex]
[latex]a)\\(frac{25}{81})^{x^{3}-frac{1}{2}}=(1,8)^{2x-x^{2}}\\((frac{5}{9})^{2})x^{3-frac{1}{2}}=(frac{9}{5})^{2x-x^{2}}\\(frac{5}{9})^{2x^{3}-1}=(frac{5}{9})^{x^{2}-2x}}\\2x^{3}-1 = x^{2}-2x\\2x^{3}-x^{2}+2x-1 = 0\\x^{2}(2x-1)+(2x+1) = 0\\(x^{2}+1)(2x-1) = 0[/latex] [latex]x^{2}+1 > 0\\2x-1 = 0\\2x = 1\\x = frac{1}{2}[/latex] [latex]b)\\(2sqrt{2}})^{x+1} = (frac{1}{4})^{-x+2}\\(2*2^{frac{1}{2}})^{x+1}}=((2^{-2))-x+2}\\2^{frac{3}{2}(x+1)}}=2^{2x-4}\\frac{3}{2}(x+1)=2x-4 |*2\\3(x+1) = 4x-8\\3x+3 = 4x-8\\3x-4x = -8-3[/latex] [latex]-x = -11\x = 11[/latex] [latex]c)\\(frac{9}{sqrt{3}})^{x} = 3^{x+1}\\(frac{9}{sqrt{3}}*frac{sqrt{3}}{sqrt{3}})^{x} = 3^{x+1}\\(3sqrt{3}})^{x} = 3^{x+1}\\(3*3^{frac{1}{2}})^{x}=3^{x+1}\\3^{frac{3}{2}x} = 3^{x+1}\\frac{3}{2}x = x+1 |*2\\3x = 2x+2[/latex] [latex]3x-2x = 2\\x = 2[/latex] [latex]d)\\(frac{81sqrt[3]{3}}{sqrt{27}})^{6x} = 3^{x^{2}}\\(3^{4+frac{1}{3}-frac{3}{2}})^{6x}=3^{x^{2}}\\(3^{frac{17}{6}})^{6x} = 3^{x^{2}}\\3^{17x} = 3^{x^{2}}\\17x = x^{2}\\17x-x^{2} = 0[/latex] [latex]x^{2}-17x = 0\\x(x-17) = 0\\x = 0 vee x = 17[/latex]
