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***Title*

*Irrational to Rational*

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***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? *

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***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?

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***Author & Contact*

Angela Gilliam, Memorial Middle School

agilliam@rockdale.k12.ga.us

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***Link(s) to resources, references, lesson plans, and/or other
materials*

Click here to view problem
at InterMath

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