[AM_20151117ISCP] Estimating a Crazy “Prosperous” surd to the Nearest Integer

Question

Introduction

     To the Chinese, the number  8  (八)  is considered to be auspicious, because it sounds like “prosper” (发) in the various Chinese languages/dialects.  But the LHS expression featured above seems too prosperous for comfort.  There is an explosion of  8s  coupled with eighth roots.  How to even handle that?

Strategy

     As usual, often one good tactic is to look for patterns or chunks.  [H04]  Can you see the sub-expressions containing the eighth roots (highlighted in green and pink)?

Do you notice any similarities between the two chunks?  Do you notice any difference(s)?

If we call the green chunk  a  and the pink chunk  b,  we can remove the roots by taking the eighth powers.  Then we get whole numbers, which are less complicated.

Solution

Remark

     The crux of the problem is the factorisation of  a⁸ – b⁸.  It is based on repeated application of the difference of squares    X² – Y² = (X + Y) (X – Y)  formula which schools expect students to know.
     For another example of using the difference of squares formula, please look at thisarticle.

Heuristics Used

H04. Look for pattern(s)

H09. Restate the problem in another way

H10. Simplify the problem

H11. Solve part of the problem

H13* Use Equation / write a Mathematical Sentence

Suitable Levels
* “IP Mathematics” so called
* challenge for students taking GCE ‘O’ Level Additional Mathematics
* other syllabuses that involve surds
* precocious kids who like to test themselves