4.
[tex]a) \ x^{-2}\cdot(x^{-3}+2x^{-2}+3x^{-3}) = x^{-2}\cdot(2x^{-2}+4x^{-3}) = 2x^{-2+(-2)} + 4x^{-2+(-3)}=\\\\=\boxed{2x^{-4}+4x^{-5}}[/tex]
[tex]b) \ 2x^{-1}\cdot(x-x^{2}+2x^{3}+3x^{-2}) = 2x^{-1+1}-2x^{-1+2} + 4x^{-1+3} + 6x^{-1+(-2)} =\\\\=2x^{0}-2x^{1}-4x^{2}+6x^{-3} = 2\cdot1-2x-4x^{2}+6x^{-3}=\boxed{2-2x+4x^{2}+6x^{-3}}[/tex]
[tex]c) \ (x-2x^{-2}\cdot(x-3x^{-1}+4x^{-2}) =x^{2}-3x^{0}+4x^{-1}-2x^{-1}+6x^{-3}-8x^{-4}=\\\\=x^{2}-3\cdot1 +2x^{-1}+6x^{-3}-8x^{-4}=\boxed{x^{2}-3+2x^{-1} +6x^{-3}-8x^{-4}}[/tex]
[tex]d) \ (2x^{-3}+4x^{-2})\cdot(x^{2}+3x^{2}+5x^{-5}+4x^{-3}+1)=\\\\=(2x^{-3}+4x^{-2})\cdot(4x^{2}+5x^{-5}+4x^{-3}+1)=\\\\=8x^{-3+2} + 10x^{-3-5} + 8x^{-3-3} + 2x^{-3} +16x^{-2+2}+20x^{-2-5}+16x^{-2-3}+4x^{-2}=\\\\=8x^{-1}+10x^{-8}+8x^{-6}+2x^{-3}+16x^{0}+20x^{-7}+16x^{-5} + 4x^{-2}=\\\\=8x^{-1}+10x^{-8}+8x^{-6}+2x^{-3}+16\cdot1+20x^{-7}+16x^{-5}+4x^{-2}=\\\\\boxed{=16+8x^{-1}+4x^{-2}+2x^{-3}+16x^{-5}+8x^{-6}+20x^{-7}+10x^{-8}}[/tex]
Wykorzystano prawa potęgowania:
[tex]a^{0} = 1 \ \ \ \ dla \ a\neq 0\\\\a^{1} = a \ \ \ \ dla \ kazdego \ a\\\\a^{m}\cdot a^{n} = a^{m+n}[/tex]
