a)
3−236=
=3−236⋅3+233+23=
=(3−23)⋅(3+23)6⋅(3+23)=
=9−126⋅(3+23)=
=−3 16 2⋅(3+23)
=−6−43
b)
23+26=
=23+26⋅23−223−2=
=(23+2)⋅(23−2)6⋅(23−2)=
=12−2218 −12=
=1022⋅9 −3⋅4=
=102⋅32−23=
=532−3
c)
1+323
Wykorzystamy wzór skróconego mnożenia dla trzeciej potęgi:
a3+b3=(a+b)(a2−ab+b2)
W naszym przypadku mamy:
a=1
b=32
a2−ab+b2=1−32+(32)2
1+323⋅1−32+(32)21−32+(32)2=
=13+(32)33(1−32+(32)2)=
=1+23(1−32+(32)2)=
=3 13 1(1− 32+(32)2)=
=1−32+ 34
d)
33−321
a3−b3=(a−b)(a2+ab+b2)
a=33
b=32
a2+ab+b2=(33)2+33⋅32+(32)2
33−321⋅(33)2+33⋅32+(32)2(33)2+33⋅32+(32)2=
=(33)3−(32)3(33)2+33⋅32+(32)2=
=3−2(33)2+33⋅32+(32)2=
=39+36+34
e)
2+3−12=
=2+(3−1)2⋅2−(3−1)2−(3−1)=
=(2)2−(3−1)22(2−(3−1))=
=2−3+23−12(2−(3−1))=
=−2 1+2 132 1(2−(3−1))=
=3−12−(3−1)⋅3+13+1=
=3−1(3+1)⋅(2−(3−1))=
=2((3+1)⋅2−(3+1)(3−1))=
=26+2−3+1=
=22+6−2