Korzystamy ze wzorów skróconego mnożenia:
[tex](a+b)^{2} = a^{2}+2ab+b^{2}\\\\(a-b)^{2} = a^{2}-2ab + b^{2}[/tex]
Oraz z własności potęg:
[tex]a^{0} = 1 \ \ \ dla \ a\neq 0\\\\a^{1} = a \ \ \ dla \ kazdego \ a\\\\(a^{n})^{m} = a^{n\cdot m}\\\\a^{n}\cdot a^{m} = a^{n+m}[/tex]
6.
[tex]a) \ (x^{-2}+x^{-3})^{2} = (x^{-2})^{2} +2\cdot x^{-2}\cdot x^{-3} + (x^{-3})^{2} =x^{-2\cdot2}+2\cdot x^{-2-3}+x^{-3\cdot2} =\\\\=\boxed{x^{-4}+2x^{-5}+x^{-6}}[/tex]
[tex]b) \ (x-3x^{-1})^{2}=x^{2}-2\cdot x\cdot3x^{-1}+(3x^{-1})^{2} =x^{2}-6x^{1-1}+3^{2}\cdot x^{-1\cdot2}=\\\\=x^{2}-6x^{0}+9x^{-2}=x^{2}-6\cdot1 + 9x^{-2} =\boxed{x^{2}-6+9x^{-2}}[/tex]
[tex]c) \ (4x^{-5}-2x^{-4})^{2} = (4x^{-5})^{2} -2\cdot4x^{-5}\cdot2x^{-4}+(2x^{-4})^{2} =\\\\=4^{2}\cdot x^{-5\cdot2}-16x^{-5-4}+2^{2}\cdot x^{-4\cdot2}=\boxed{16x^{-10}-16x^{-9}+4x^{-8}}[/tex]
[tex]d) \ (3x^{-2}+x^{-1})^{2}=(3x^{-2})^{2} + 2\cdot3x^{-2}\cdot x^{-1} + (x^{-1})^{2} =\\\\=3^{2}\cdot x^{-2\cdot2} +6x^{-2-1} + x^{-1\cdot2} = \boxed{9x^{-4}+6x^{-3}+x^{-2}}[/tex]
[tex]e) \ (2x-x^{-4})^{2} = (2x)^{2} -2\cdot2x\cdot x^{-4}+(x^{-4})^{2} = 2^{2}\cdot x^{2}-4x^{1-4}+x^{-4\cdot2}=\\\\=\boxed{4x^{2}-4x^{-3}+x^{-8}}[/tex]
[tex]f) \ (5x^{-1}+2x^{-3})^{2}=(5x^{-1})^{2}+2\cdot5x^{-1}\cdot2x^{-3}+(2x^{-3})^{2} =\\\\=5^{2}\cdot x^{-1\cdot2} +20x^{-1-3} +2^{2}\cdot x^{-3\cdot2} = \boxed{25x^{-2}+20x^{-4}+4x^{-6}}[/tex]
