A. (√2/2 - √3/2) * (√2/2 + √3/2)=(√2/2)²-(√3/2)²=2/4-¾=-¼ B. (3*√2/2 + √3) * (3*√2/2 - √3)=(3*√2/2)² - (√3)=3/4-3=-2¼ C. (√3/2 + √3/2)² - (1/2 + 1/2)²=(√3/2)²-1²=¾-1=-¼ +D. (√3 - 1/2) * (1/2 - √3)= (√3 - 1/2) * (-)(√3+½)=-[(√3)²-½²]=-(3-¼=-3+¼=-2¾ E. 4(1+√3/2) * (√3/2 + 1)=4(√3/2 + 1+¾+√3/2)=4(2√3/2+1¾)=4(√3+1¾)=4√3+7
We wszystkich skorzystac trzeba ze wzorów skróconego mnożenia (a-b)(a+b)=a²-b² a. (√2/2 - √3/2) * (√2/2 + √3/2)=(√2/2)²-(√3/2)²=2/4-3/4=-1/4 B. (3*√2/2 + √3) * (3*√2/2 - √3) =18/4-3=4,5-3=1,5 (a+b)²=a²+2ab+b² C. (√3/2 + √3/2)² - (1/2 + 1/2)²= (√3/2)²-2*(√3*√3)/(2*2)- (1)²=3/4-6/4-1=-3/4-1=-7/4 D. (√3 - 1/2) * (1/2 - √3)=√3*1/2-√3*√3-1/2*1/2+√3*1/2=2√3/2-3-1/4= √3 -3-1/4=-3 1/4- √3 E. 4(1+√3/2) * (√3/2 + 1) 4*[√3 /2+1+3/4+√3 /2]=4*[1 3/4+2√3 /2]=4*[7/4+√3 ]=7+4√3
Korzystamy tu ze wzorów skróconego mnożenia: (a-b)*(a+b)=a²-b² oraz (a+b)²=a²+2ab+b² A (√2/2 - √3/2)*(√2/2 + √3/2) = ²/₄ - ¾ = -¼, bo √2*√2=2 i 2*2=4, √3*√3=3 stąd te liczby B (3*√2/2 + √3) * (3*√2/2 - √3) = 9*2/4 - 3 = 18/4 - 3 = 4½ - 3 = 1½, bo (3*√2)²=9*2=18 C (√3/2 + √3/2)² - (1/2 + 1/2)² = (2√3/4)² - 1 = 4*3/16 - 1 = 12/16 - 1 = -¼ D (√3 - 1/2) * (1/2 - √3) = ½√3 - 3 - ¼ + ½√3 = √3 - 3¼ E 4(1+√3/2) * (√3/2 + 1) = 4(√3/2 + 1 + ¾ +√3/2) = 2√3 + 4 + 3 + 2√3 = 4√3 + 7 Wszystko jasne?? :)
