趣味數學一條(20分)

Let x = √(7+√13)-√(7-√13)

(Remarks: x^2 ==＞ x square / x 二次方）
Therefore
x^2 = (√(7+√13)-√(7-√13))^2
= (7+√13) + (7-√13) - 2 (√(7+√13)) (√(7-√13))
= 7 + √13 + 7 -√13 - 2 (√(49 - 13))
= 14 - 2 √36
= 14 - 2 * 6
= 2

Therefore x = √2

GOOD!!! 加油!!!

Sorry to copy from others for part of the reply first of all, the effort shown at the bottom should the best among several answers before. By instinct it is expected to have one and only one solution only. However by substitution and squaring the whole term, simplifying and finding roots, two roots should come from the square of the whole term. In order to obtain one answer by claiming the term √(7+√13)-√(7-√13) not smaller than 0 without proof or by just ignoring the negative solution, is not quite satisfying. There may be a way to escape squaring the term that most of us do not know, but proving √(7+√13)-√(7-√13) not smaller than than 0 should not be less significant than squaring the term. I shall expect a treatment like this:

As 7>0, √13>0,
(7+√13)>(7-√13)>0
√(7+√13)>√(7-√13)>0
thus
√(7+√13)-√(7-√13)>0

Quite uneasy way to eliminate one answer, but any better way?

The followings are to show what had been done and appreciate for effort.

==
let A = √(7+√13)-√(7-√13)
and

A^2
= [√(7+√13)-√(7-√13)]^2
= (7+√13) -2[ √(7+√13) ][ √(7-√13)] + (7-√13)
= 14 - 2[ √(49 -13) ]
= 14 - 12
= 2

and
A = +/-√2

and so, √(7+√13)-√(7-√13)
= +/-√2
the answer, which is equal to -√2, is rejected, since A 0

so the answer should be √2
==

2006-10-20 12:30:59 補充：
Acknowledgement: mafanr for the section copiedNote: > is the larger than sign

2006-10-21 22:17:41 補充：
答案庸，選擇更庸。
不知答下面一題會否一樣？
√(7-√13) - √(7+√13)

let A = √(7+√13)-√(7-√13)
Consider A^2 (because we want to remove the root sign)
= [√(7+√13)-√(7-√13)]^2
= (7+√13) -2[ √(7+√13) ][ √(7-√13)] + (7-√13)
= 14 - 2[ √(49 -13) ]
= 14 - 12
= 2

thus, A = √2 (reject the negative value because the root should be positive)

Let x = √(7+√13)-√(7-√13)

(Remarks: x^2 ==＞ x square / x 二次方）
Therefore
x^2 = (√(7+√13)-√(7-√13))^2
= (7+√13) + (7-√13) - 2 (√(7+√13)) (√(7-√13))
= 7 + √13 + 7 -√13 - 2 (√(49 - 13))
= 14 - 2 √36
= 14 - 2 * 6
= 2

Therefore x = √2

14(1-√13)