Wyznacz: x+ y, x-y, x razy y, x/y, x do kwadratu (wynik doprowadz do postaci a+b pierwiastek z c), gdy:
a)
x=3+2\sqrt3 , y=2-\sqrt3
x+y=3+2\sqrt3+2-\sqrt3=5+\sqrt3
x-y=3+2\sqrt3-(2-\sqrt3)=3+2\sqrt3-2+\sqrt3=1+3\sqrt3
x*y=(3+2\sqrt3)(2-\sqrt3)=6-3\sqrt3+4\sqrt3-2*3=\sqrt3
\frac{x}{y}=\frac{3+2\sqrt3}{2-\sqrt3}=\frac{(3+2\sqrt3)(2+\sqrt3)}{(2-\sqrt3)(2+\sqrt3)}=\frac{6+3\sqrt3+4\sqrt3+2*3}{4-3}=12+7\sqrt3
b)
x=2-5\sqrt7, y=1-\sqrt7
x+y=2-5\sqrt7+1-\sqrt7=3-6\sqrt7
x-y=2-5\sqrt7-(1-\sqrt7)=2-5\sqrt7-1+\sqrt7=1-4\sqrt7
x*y=(2-5\sqrt7)(1-\sqrt7)=2-2\sqrt7-5\sqrt7+5*7=37-7\sqrt7
x^2=(2-5\sqrt7)^2=4-20\sqrt7+25*7=179-20\sqrt7
c)
x=4+5\sqrt2, y=3+3\sqrt2
x+y=4+5\sqrt2+3+3\sqrt2=7+8\sqrt2
x-y=4+5\sqrt2-(3+3\sqrt2)=4+5\sqrt2-3-3\sqrt2=1+2\sqrt2
x*y=(4+5\sqrt2)(3+3\sqrt2)=12+12\sqrt2+15\sqrt2+15*2=42+27\sqrt2
\frac{x}{y}=\frac{4+5\sqrt2}{3+3\sqrt2}=\frac{(4+5\sqrt2)(3-3\sqrt2)}{(3+3\sqrt2)(3-3\sqrt2)}=\frac{12-12\sqrt2+15\sqrt2-15*2}{9-9*2}=\frac{-18+3\sqrt2}{-9}=\frac{-3(6-\sqrt2)}{-9}=\frac{6-\sqrt2}{3}
x^2=(4+5\sqrt2)^2=16+40\sqrt2+25*2=66+40\sqrt2