Square Roots - Part III (Factoring Square Roots)

Where I left off in my previous post about irrational numbers, we were trying to solve for the square root of 24 by trial and error.  Reducing an irrational number can be a tedious job if you do it this way!  Luckily, unless specified otherwise, you are allowed to leave your answer in the "simplified radical form," which you can get by factoring square roots.

"Simplified radical form" is exactly what it sounds like: you simplify your expression and leave it expressed as some radical.  But, you have a little bit of work to do to reduce it.

Factoring square roots is quite simple.  It has to do with factoring perfect squares.  But first, you have to determine the factors of the number under the radical sign, and then if any of those factors are perfect squares, you can pull it (the square root of the perfect square factor) out from underneath the radical sign and put it in front to multiply by it.  That sounds awfully wordy and probably isn't the most concise definition, but I think an example will go a long way to helping you understand factoring square roots, and then leaving them in simplified radical form.

Let's continue with the example from my last post, looking for the square root of 24.  In this case, let's just reduce it to its simplified radical form, and not bother wasting time trying to find the exact decimal answer.

So, the first step is to ask yourself what the factors of 24 are.  (I'm going to ignore the √ for a minute)  For this, you can determine this quite easily by trial and error.  You can find that the factors of 24 are 1,2,3,4,6,8,12,24.  From these now, you want to see if any are perfect squares.  Again, you should be able to easily say that 4 is the perfect square.  So, 24 can be expressed as 4 x 6. That's what we want.  Now, let's look at what we have done so far:

√24 = √(4x6)

Now, as I said before, we can take the perfect square that is under the radical sign, and bring it outside the sign.  This is because of the properties of square roots (property #2), which in this case allows us to write:

√(4x6) = √4 x √6

With that property in mind, it should be a bit easier to see why we are interested in the perfect squares, because now, by factoring perfect squares, we can just rewrite the square root of 4 to 2!  So, we can finally write our expression in simplified radical form as:

√24 = 2√6

I hope this demonstration has explained to you the basics of factoring square roots, and leaving you answers in the simplified radical form.  Let me know in the comments if you'd like another example, and I'll do one for you!  Remember to +1 me if this helped you!  :)

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