6) Rozwiąż równanie: [latex] 25^{9}x= frac{5^{17}- 5^{16}}{125^{2}*25^{ -frac{9}{2}} } [/latex] 7) Dana jest funkcja f określona wzorem: f(x)=[latex]( frac{1}{4})^{x} [/latex] Oblicz: [latex]f(frac{3}{2})} - f (-2)[/latex]

[latex]25^{9}x=frac{5^{17}-5^{16}}{125^{2}*25^{-frac{9}{2}}}\ \ (5^{2})^{9}x=frac{5^{16}(5-1)}{(5^{3})^{2}*(5^{2})^{-frac{9}{2}}}\ \ 5^{2*9}x=frac{5^{16}*4}{5^{3*2}*5^{-frac{9}{2}*2}}\ \ 5^{18}x=frac{5^{16}*4}{5^{6}*5^{-9}}\ \ 5^{18}x=frac{5^{16}*4}{5^{6-9}}\ \ 5^{18}x=5^{16}*4:5^{-3}\ \ 5^{18}x=5^{16-(-3)}*4\ \ 5^{18}x=5^{19}*4 |:5^{18}\ \ x=5^{19-1}*4\ \ x=5*4\ \ x=20[/latex] ====================================================== [latex]f(x)=(frac{1}{4})^{x}=((frac{1}{2})^{2})^{x}=(frac{1}{2})^{2x}\ \ f(frac{3}{2})=(frac{1}{2})^{2*frac{3}{2}}=(frac{1}{2})^{3}=frac{1}{8}\ \ f(-2)=(frac{1}{2})^{2*(-2)}=(frac{1}{2})^{-4}=2^{4}=16\ \ f(frac{3}{2})-f(-2)=frac{1}{8}-16=-15frac{7}{8}[/latex]