a)
(\frac{2}{3})^{x^2}=(1\frac{1}{2})^{2x^2+x}
(\frac{2}{3})^{x^2}=(\frac{3}{2})^{2x^2+x}
(\frac{2}{3})^{x^2}=(\frac{2}{3})^{-(2x^2+x)}
x^2=-(2x^2+x)
x^2=-2x^2-x
x^2+2x^2+x=0
3x^2+x=0
x(3x+1)=0
x=0\vee 3x=-1
x=0\vee x=-\frac{1}{3}
b)
2^{x^3+x^2}=32^{x+1}
2^{x^3+x^2}=2^{5(x+1)}
x^3+x^2=5(x+1)
x^3+x^2=5x+5
x^3+x^2-5x-5=0
x^2(x+1)-5(x+1)=0
(x+1)(x^2-5)=0
(x+1)[x^2-(\sqrt5)^2]=0
(x+1)(x-\sqrt5)(x+\sqrt5)=0
x=-1\vee x=\sqrt5\vee x=-\sqrt5
c)
(\frac{25}{81})^{x^3-\frac{1}{2}}=(1,8)^{2x-x^2}
(\frac{5}{9})^{2(x^3-\frac{1}{2})}=(\frac{18}{10})^{2x-x^2}
(\frac{5}{9})^{2x^3-1}=(\frac{9}{5})^{2x-x^2}
(\frac{5}{9})^{2x^3-1}=(\frac{5}{9})^{-(2x-x^2)}
2x^3-1=-(2x-x^2)
2x^3-1=x^2-2x
2x^3-x^2+2x-1=0
x^2(2x-1)+(2x-1)=0
(2x-1)(x^2+1)=0 …x^2+1>0
dla każdej liczby \mathbb R
2x-1=0
2x=1
x=\frac{1}{2}
d)
\frac{3^{x^3}}{81}=9^{x-x^2}
\frac{3^{x^3}}{3^4}=(3^2)^{x-x^2}
3^{x^3-4}=3^{-2x^2+2x}
x^3-4=-2x^2+2x
x^3+2x^2-2x-4=0
x^2(x+2)-2(x+2)=0
(x+2)(x^2-2)=0
(x+2)(x^2-(\sqrt2)^2)=0
(x+2)(x-\sqrt2)(x+\sqrt2)=0
x=-2\vee x=\sqrt2\vee x=-\sqrt2