快快快!聽日要交功課啦!!二次根式!!

1 1/(1+√2)+1/(√2+√3)+1/(√3+√4)+1/(√4+√5)+⋯+1/(√99+√100)

= (√2 - 1) + (√3 - √2) + (√4 - √3)+...+( √100 - √99)

= √100 - 1 = 9

2 (x+y+2√xy)/(√x+√y)+(x-y)/(2√x-2√y)

= [(√x+√y)^2]/(√x+√y) + (√x+√y)(√x-√y) / 2(√x-√y)

= (√x+√y) + (√x+√y)/2

= 3(√x+√y) / 2

3求證 若a=√2+1 求證√2-1= 1/(2+1/(2+1/(2+1/(2+1/a))))

2 + 1 /a = 2 + 1/(√2+1) = 2 + (√2 - 1) = √2 + 1 = a , so:

1/(2+1/(2+1/(2+1/(2+1/a ))))

= 1/(2+1/(2+1/(2+1/(a)))

= 1/(2+1/(2+1/a))

= 1/(2+1/a)

= 1/a

= 1/(√2+1) = =√2 - 1

4) 求證 若a,b,c,d為有理數,且a+b√2=c+d√2 求證a=c b=d


(a-c) + (b-d)√2 = 0

(b-d)√2 = (c-a)

√2 = (c-a)/(b-d)

若 b 不 = d

√2 = M/N = 有理數,茅盾

所以 b = d = k

a+k√2=c+k√2

a = c

1 1/(1+√2)+1/(√2+√3)+1/(√3+√4)+1/(√4+√5)+⋯+1/(√99+√100)


2 (x+y+2√xy)/(√x+√y)+(x-y)/(2√x-2√y)


3求證 若a=√2+1 求證√2-1= 1/(2+1/(2+1/(2+1/(2+1/a))))


4求證 若a,b,c,d為有理數,且a+b√2=c+d√2 求證a=c b=d