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 Write-up

Title
Irrational to Rational

Problem Statement
When will the product of two square roots result in an integer if each of the individual square roots is not equal to an integer?

Problem setup

Essentially, we want to find out the cases in which the product of two square roots, that have rational parts or parts that do not result in perfect squares, will result in an integer solution.  I must investigate the possible cases that would result in the elimination of the root symbol in this problem.

Plans to Solve/Investigate the Problem

What information do I know about multiplying square roots that will help me solve this problem?  The product of square roots will result in an integer in a few cases.  First, if the square root is a perfect square, the answer will result in an integer.  Secondly, if the product of two square roots result in the formation of a perfect square under the radical sign, the answer will result in an integer.  Further, if the product of two negative square roots result in a positive perfect square, the result would be an integer.

Investigation/Exploration of the Problem

I first considered the case in which a square root that does not result in an integer is squared; for example, (√18)2, the result would be the integer 18.  √18  =  3√2.  (3√2)2 = 9 x 2 = 18.

Secondly, I considered the case in which the product of the radical portion of two square roots results in a perfect square.  For example, the product of 4√3  x  3√3.  This problem is simplified as the product of 4  x  3 x  √3 x  √3 =  12  x  √9  =  12  x 3  = 36.

In the case of negative square roots, √-1  x  √-4  = √ 4  = 2.

Extensions of the Problem

What other rational numbers could you consider (i.e., п, e, etc.)?  What result might you obtain in these cases?

Author & Contact
Angela Gilliam, Memorial Middle School
agilliam@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
Click here to view problem at InterMath

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