[latex]a)\\(3sqrt{2}-4)(3sqrt{2}+4)= (3sqrt{2})^2-4^2=9*2-16=18-16=2\\b)\\(3+2sqrt{5} )(3-2sqrt{5} )= 3^2-( 2sqrt{5} )^2=9-4*5=9-20=-11[/latex] [latex]c)\\ (2+sqrt{3} )^2(7-4sqrt{3} )= (4+2*2*sqrt{3}+(sqrt{3})^2)(7-4sqrt{3} )=\\=(4+4sqrt{3}+3)(7-4sqrt{3} )=(7+4sqrt{3} )(7-4sqrt{3} )=7^2-(4sqrt{3})^2=\\=49-16*3=49-48=1[/latex] [latex]d)\\ (1-2sqrt{2} )^2(9+4sqrt{2} )= (1^2-2*1*2sqrt{2}+(2sqrt{2})^2 ) (9+4sqrt{2} )=\\=(1 -4sqrt{2}+4*2 ) (9+4sqrt{2} )=(1 -4sqrt{2}+ 8) (9+4sqrt{2} )=\\=(9 -4sqrt{2} ) (9+4sqrt{2} )=9^2-(4sqrt{2})^2=81-16*2=81-32=49[/latex] [latex]e)\\ (3+sqrt{7} )^2-(3-sqrt{7} )^2=(3+sqrt{7}-(3-sqrt{7}))(3+sqrt{7}+ 3-sqrt{7} )=\\=(3+sqrt{7}- 3+sqrt{7} )(3+sqrt{7}+ 3-sqrt{7} )=2sqrt{7}*6=12sqrt{7}[/latex] [latex]f)\\(2sqrt{2}-sqrt{3} )^2+ (2sqrt{3}+sqrt{2} ) ^2=\\=(2sqrt{2})^2-2*2sqrt{2}*sqrt{3} +(sqrt{3})^2+ (2sqrt{3})^2+2*2sqrt{3}*sqrt{2} +(sqrt{2})^2=\\=4*2 -4sqrt{3*2} +3+ 4*3+4sqrt{3*2} +2=8 -4sqrt{6} +3+12+4sqrt{6} +2=25[/latex] [latex](a-b)(a+b)=a^2-b^2\\( a-b)^2=a^2-2ab+b^2\\( a+b)^2=a^2+2ab+b^2[/latex]
Należy użyć wzorów skróconego mnożenia: Kwadrat sumy (różnicy):[latex](apm b)^2=a^2pm2ab+b^2[/latex] Różnica kwadratów:[latex](a-b)(a+b)=a^2-b^2[/latex] [latex]a)\(3sqrt2-4)(3sqrt2+4)=(3sqrt2)^2-4^2[/latex] skorzystamy z praw działań na potęgach i pierwiastkach: [latex](acdot b)^n=a^ncdot b^n i (sqrt{a})^2=a dla a extgreater 0[/latex] [latex]=3^2(sqrt2)^2-16=9cdot2-16=18-16=oxed{2}[/latex] [latex]b)\(3+2sqrt5)(3-2sqrt5)=3^2-(2sqrt5)^2=9-2^2cdot(sqrt5)^2=9-4cdot5\\=9-20=oxed{-11}[/latex] [latex]c)\(2+sqrt3)^2(7-4sqrt3)=[2^2+2cdot2cdotsqrt3+(sqrt3)^2]cdot(7-4sqrt3)\\=(4+4sqrt3+3)cdot(7-4sqrt3)=(7+4sqrt3)(7-4sqrt3)\\=7^2-(4sqrt3)^2=49-4^2cdot(sqrt3)^2=49-16cdot3=49-48=oxed{1}[/latex] [latex]d)\(1-2sqrt2)^2(9+4sqrt2)=[1^2-2cdot1cdot2sqrt2+(2sqrt2)^2]cdot(9+4sqrt2)\\=[1-4sqrt2+2^2cdot(sqrt2)^2]cdot(9+4sqrt2)=(1-4sqrt2+4cdot2)cdot(9+4sqrt2)\\=(1-4sqrt2+8)cdot(9+4sqrt2)=(9-4sqrt2)(9+4sqrt2)=9^2-(4sqrt2)^2\\=81-4^2cdot(sqrt2)^2=81-16cdot2=81-32=oxed{49}[/latex] [latex]e)\(3+sqrt7)^2-(3-sqrt7)^2=[(3+sqrt7)-(3-sqrt7)]cdot[(3+sqrt7)+(3-sqrt7)]\\=(3+sqrt7-3+sqrt7)cdot(3+sqrt7+3-sqrt7)=2sqrt7cdot6=oxed{12sqrt7}[/latex] [latex]f)\(2sqrt2-sqrt3)^2+(2sqrt3+sqrt2)^2=(2sqrt2)^2-2cdot2sqrt2cdotsqrt3+(sqrt3)^2\\+(2sqrt3)^2+2cdot2sqrt3cdotsqrt2+(sqrt2)^2\\=2^2cdot(sqrt2)^2-4sqrt{2cdot3}+3+2^2cdot(sqrt3)^2+4sqrt{3cdot2}+2\\=4cdot2-4sqrt6+3+4cdot3+4sqrt6+2\\=8+3+12+2=oxed{25}[/latex]
