At the first glance it seems rather simple problem to solve, only to discover how fast we have lost touch with the basics of indices and algebra. The problem could be simple for most but definitely is difficult for me during the first few hours.
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Think and think, try and try .... but no progress. Could be thinking of the wrong thing!
When the first few preliminary attempts by just using the “brain” failed to produce any presentable answer, there is doubt whether I got any (brain).
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I proceed to hunt for relevant text books. Unfortunately the available books do not have similar examples. I dig into some of my younger days, almost lost and forgotten school books, also without any results. My ego is beginning to get bruised. What is assumed to be easy and simple has turn out to be so complicated.
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There are many representations of the same thing. Miraculously we can still manage to understand and remember.
This picture is obtained from http://mathsteaching.files.wordpress.com/2008/01/fractional-powers-03.jpg
Those are meant for primary students? I hope they have additional guidance. This is probably why full-time teachers are necessary. Those that insist on "teach less to learn more" may not work.
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It is really no joke. Student need to put in a lot of efforts by doing and doing similar problems. The comment "If you don't help yourself, who will?" is useful here.
Feeling quite defeated, I try using the Internet. There lies tremendous amount of information and theories which long time ago is already given back to the source. After one to two hours of learning the many basics, there is still no clue on how to solve the question.
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Looking and searching very hard, but lady luck is just not smiling.
The internet suggests Arithmetic Sequence Number, Fibonacci number, Arithmetic and Geometric Progressions, Fourier Series, Binomial Theorem and Pascal’s Triangle. These make us realized why such term PHD (Permanent Head Damage?) existed.
The old machine is ready to give up and call it a day.
Then some conversation is heard on the phone’s speaker. My nephew has given up on me and is calling his classmate for help. A few moments of “Yes, yes …”, “I see …”, “That one hah ...”, and then “Oh! I know already!”
Within a while he found the solution with the valuable help from his "Gu Gu".
Using algebraic expanding,
We have, for an expression,
Performing the same step for the whole expression,
We will get ,
Simplifying by canceling similar nominator and denominator,
Simplifying further,
= 7
Therefore the expression,
The method used is not complicated but tedious. There could be other easier methods which we hope to discover.
What is already discovered is this “Climbing the wrong ladder” is disastrous. Our stubborn inclination to use “Advanced” algebra is wasting effort and time when a simple expanding and contracting (simplifying) is enough.
利 用 牛 刀 杀 鸡 － “Use a big chopper to kill a chicken”
Lì yòng niú dāo shā jī
Summary of Index Laws
The laws are obtained from http://www.intmath.com/Numbers/exp.gif
What is remain to be discovered is how the “advanced“ mathematical theorems and the different sequences of number can help us in our life. Many of us have not yet apply them successfully. They are purely for academic purpose.
Who is able to show us?
Maybe then we can really celebrate with the "LOVE IS ALL".
This picture is obtained from http://www.dio.net/pictures_cd/the_butterfly_ball_fr_pomme_big.jpg
For almost half a decade, all from Philips' DVD Audio and Video department have come to know about this disc.
It is a daily affair to playback the songs to test the reliability of the systems is according to the specifications stated in the PQR (Performance, Quality and Reliability) manual.
This picture is obtained from http://www.dio.net/pictures_cd/the_butterfly_ball_front_big.jpg
Most Philips' testers and ex-colleague will recognize and remember this songs playback from DVD disc LVP 04.15 T6, T14, T22, T30, T38, T47 and T55. 
This disc is meticulously designed to captured the problems of Payers, Home Theater System, Home Theater Receiver and Recorder.
The design engineers have utilized the advanced algebraic techniques while designing this master test disc.
All is not lost, many fond memories still remain.
Reference
http://www.math.uow.edu.au/subjects/summer/refresh.html http://mathsteaching.files.wordpress.com/2008/01/fractional-powers-02.jpg
http://mathsteaching.files.wordpress.com/2008/01/fractional-powers-03.jpg
http://mathsteaching.files.wordpress.com/2008/01/fractional-powers-04.jpg
http://www.intmath.com/Numbers/exp.gif
http://www.dio.net/pictures_cd/the_butterfly_ball_front_big.jpg
http://www.dio.net/pictures_cd/the_butterfly_ball_fr_pomme_big.jpghttp://wii.kombo.com/images/content/news/600px-MKWii_Blueshell.jpg
http://www.phillyscreenwriter.com/wp-content/uploads/2007/05/scaredguy.jpg
http://www.pugwash.org/uk/images_uk/Thinker.gif
http://llamapeopleclothing.com/thinker.jpg
Philips colleagues, especially those out sourced