[AM_20150412SQS] Perfect Squares Lurking Absurdly

Question

Introduction

     Surds are expressions involving roots like square roots.  They are usually irrational numbers.  If you try to put them into a ratio of integers, you are absurd!  Many students (and teachers?) are not sure of how to put square roots of numbers in a simple form.  The trick is to use square numbers or perfect squares.  These are squares of whole numbers.  For example:-

     1² = 1 ´ 1 = 1                      Ö1 = 1

     2² = 2 ´ 2 = 4                      Ö4 = 2

     3² = 3 ´ 3 = 9                      Ö9 = 3

     4² = 4 ´ 4 = 16                  Ö16 = 4

     5² = 5 ´ 5 = 25                  Ö25 = 5

A number like  4  can be represented by a real square whose sides have length  2  units.  Note also that if you take the square root of a perfect square, you always get a nice whole number.

     How do you deal with numbers that are not perfect squares?  You factor out as many perfect squares as possible.  This would eventually lead to surds with small numbers, which are more manageable.  Here are some examples:-

With this weapon in our hands, let us kick some butt.

Solution

Moral of the Story

     Using perfect squares reduces your square roots to surds involving square roots of prime numbers, which are easier to combine or cancel.  This gives a short and sweet solution.  In mathematics, always try to do things by the cleanest way (if you can).

Heuristics Used

H10. Simplify the problem

H11. Solve part of the problem

Suitable Levels
* GCE ‘O’ Level Additional Mathematics
* revision for GCE ‘A’ Level H2 Mathematics
* revision for IB Mathematics HL / SL
* other syllabuses that involve surds
* precocious kids who always want to learn more