Quod Erat Demonstrandum

2008/02/28

Evaluate cube roots

Filed under: Additional / Applied Mathematics,HKCEE,Junior Form Mathematics — johnmayhk @ 4:02 下午

A F.5 student, Yan, asked me to do the following without using calculator

\sqrt[3]{10 + 6\sqrt{3}} + \sqrt[3]{10 - 6\sqrt{3}}

As I guessed, he was playing with cubic equations. The first idea came to my mind was, if we CAN evaluate that stuff by bare hands, the numbers 10 + 6\sqrt{3}, 10 - 6\sqrt{3} should be perfect cube. Then I tried

(1 + \sqrt{3})^3
= 1 + 3\sqrt{3} + 3(3) + 3\sqrt{3}
= 10 + 6\sqrt{3}

How lucky I was! Then I gave the solution as follows

\sqrt[3]{10 + 6\sqrt{3}} + \sqrt[3]{10 - 6\sqrt{3}}
= \sqrt[3]{(1 + \sqrt{3})^3} + \sqrt[3]{(1 - \sqrt{3})^3}
= 1 + \sqrt{3} + 1 - \sqrt{3}
= 2

In the old syllabus of additional mathematics, students were required to know how to evaluate the following

\sqrt{\sqrt{a} + \sqrt{b}}.

Nothing special indeed.

Just for further discussion, if the value of x gets near to zero,

\sqrt[3]{1 + x} - \sqrt[3]{1 - x} \approx \frac{2x}{3} – – – – – – (*)

e.g.

\sqrt[3]{1 + 0.0001} - \sqrt[3]{1 - 0.0001} = 0.000066666666790123457393689990082813... (by 小算盤)

while

\frac{2\times0.0001}{3} = 0.00006666666666666666666666666666666...

Umm, the approximation is not bad, right?

How to get the approximation like (*), believe me, it is extremely easy to create something like that by applying binomial theorem (discussed in applied mathematics (II) or mathematics and statistics).

[OT] Here’s a question for F.2 or F.5 boys. See which form will obtain the correct answer first?

Factorize (x + y)^5 - x^5 - y^5.

8 則迴響 »

  1. wordpress這個東西真的蠻好用,感謝John Sir了。

    btw, john Sir, 其實我有個問題想問你好耐了,
    其實你點睇哲學、數學同埋基督教?
    有冇果種計得越多數,接觸得越多科學,便有種使靈命信心受動搖的感覺??

    迴響 由 wonghon — 2008/02/28 @ 11:51 下午 | 回覆

  2. 是呀,看你的 xanga 也知道你用 wp 來寫日記,那麼我們可以有更多的交流了。

    本來想睡,但見你這個問題,我也安靜心神一會,『惺忪』地回應一下。

    我前教會有個弟兄兼朋友,我和他都是差不多時間進到教會。他長於電腦,我就頗愛數理。在教會中,我們也很快擔當起導師或團長的角色,教友也視我們為『兩兄弟』。及後,類似地,我們成長,建立家室。

    今天,他已在神學院接受裝備,我卻在信仰上每下愈況。我離開舊教會已一段時間,早前他一家來探望我家,我知道他仍很想關心我,只是我的『太極功夫』了得,沒有很深入的討論問題,時間也不允許。只是,他說了『一個個階段』這話讓心中反思:恐怕我真的走過(或錯過)了一個階段,走著另一條路。

    聞說一流的科學家偏向相信有神,聞說頂尖的數學家因看出宇宙的數學性偏向相信上帝是數學家;我既非一流亦非頂尖,但在醞釀信主的那段時光,我真的天真地以為:信仰可以建基在科學數學和邏輯上,還幼稚地告訴傳道人,我可以由神的全能推論神的唯一,由集合的互為從屬推導聖父聖子同等云云。愛主信徒的見證和『反基』的言論在網絡內在生活中也份量相當,問題是你以什麼眼光來看或選擇逃避哪一方的立場。有時逃避衝擊可保『靈命』,但有時可能連自己內心也欺騙了。

    我真的被愛主信徒身體力行的見證深深打動,但我也承認,我對『反基』的提問質疑也沒有答辯的把握,尤其他們的聖經/神學/科學/哲學/社會學等等的知識比你強很多很多的時候,我們不能一概以『奧秘』、「世上的小學」無視之。把信仰生活化約到困在四面牆內『你好我好』般聚會,是天真的逃避,但我卻沒有資格高談什麼,因為我也根本沒有勇氣『活出去』。

    那天,那朋友和我談什麼沒有對錯的『後現代』,我也被社會同化到有點立場不明(看到這裡你也感到吧),那我的信心有動搖吧?常有,自九一年信主至今,尤其遇上義人受難的時候,但以下聖經句語常常提我:

    『你要保守你心,勝過保守一切』

    理性告訴我,相對宇宙,人很渺小,相信我們無法完全了解『終極問題』。但了解『終極問題』的難度,遠高於教導螞蟻學懂微積分,而且,為何要螞蟻學微積分?如果信仰等於深奧的神學,和現世的生活無關,那為何還要信?大概我們這些平凡的血身之軀,透過自己的內心,觸摸過一些比知識、耍樂更崇高更美的價值,所以仍然持守著信仰的根本。至於能否『向上結果』,還是到見主面的時候才蓋棺定論了。

    為你為我禱告吧。

    迴響 由 johnmayhk — 2008/02/29 @ 1:59 上午 | 回覆

  3. =諗左兩分鐘… => 0

    迴響 由 Ed — 2008/03/02 @ 8:08 上午 | 回覆

  4. Edmund, how about after giving you that

    (x + y)^5 \equiv x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 + 5xy^4 + y^5

    迴響 由 johnmayhk — 2008/03/03 @ 12:38 上午 | 回覆

  5. ok. another fomular…
    there left the four middle terms while you can take out the common factory 5xy, but is that all?! (you are triky!)

    迴響 由 Ed — 2008/03/04 @ 7:16 下午 | 回覆

  6. Good Ed!

    The term ‘common factory’ is really awesome!

    Let me create a statement with a bit ‘mathematics feeling’.

    “We all have common factors because we are all molded in common factories: the school."

    迴響 由 johnmayhk — 2008/03/04 @ 9:50 下午 | 回覆

  7. Is the ans: 5xy(x+y)(x^2+xy+y^2)?
    Am I right?
    Also, I would like to ask the it works
    HKCEE:
    1997 paper2 Q11a
    1997 paper2 Q6,9
    1999 paper2 Q2

    what’s the method ‘substitution?

    迴響 由 harrison — 2008/03/20 @ 8:46 下午 | 回覆

  8. HONG KONGESE SLOGAN :
    " I THINK , I CHANGE , I SEE THE FUTURE !"
    MARK K. C. MA 25-12-2008 H.K.
    THE SIX DIMENSIONS OF THE UNIVERSE :
    " HEIGHT , LENGTH , WIDTH , DISTANCE , TIME AND DEPTH !"

    迴響 由 markkcma — 2008/12/24 @ 5:19 下午 | 回覆


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