There are 3 main rules for simplifying a square root.  They are:

1)  Take out all perfect squares in the number

2)  Don't leave a fraction in the square root...separate it

3)  Don't leave a radical in the denominator

In this article, we will examine each of these three rules by looking at several examples of each.  Lets first start by defining what a perfect square is.  All perfect squares can be found by squaring all of the whole numbers : 0, 1, 2 , 3, 4, 5, 6, 7, etc.   When we square these numbers we get:  0,1, 4, 9, 16, 25, 36, 49, etc.  These numbers are what we will refer to as perfect squares.

So now lets examine how to use rule #1...taking out perfect squares....by looking at this example :

√45.   I use a story to help students remember what to look for.  The radical symbol is a jail cell and the number 45 is in the jail cell.  The only way any numbers can "escape" from the cell is if the original number can be broken down into pairs of factors.  Only factors that are in pairs can escape, while factors that aren't pairs don't have a chance of escaping and must stay in jail forever.  Again, looking at the example, the number 45 can be broken down into 3*3*5.  There are a pair of 3's and then a 5 that doesn't have a partner.

The story continues and is a little sad in that while the pair of numbers attempt to "escape"...one of the partners of the pair succeeds in escaping, while the other partner gives his life for the one that succeeded. Going back to the the example above...one 3 escapes, while the other 3 gets "killed" in the process.  If there are two pairs...one of each pair escapes and the other "dies".   The two that escape get lost in the crowd so they can't get spotted so they get multiplied together with whatever might be out there already.  Further examples will show this.  The 5 doesn't have a partner, so it is doomed to stay in jail forever.  Therefore your answer this time  is 3√5.  It sounds a little violent, but the story kind of helps those students that forget which number goes on the outside of the radical sign and which ones stay inside.

Lets do one more example together before you try.  Take the example :√72.  The number 72 can be broken down into...2*2*2*3*3.  This time there are two 2's and two 3's and one 2 that does not have a partner.  One of the 2's and one of the 3's "die" and the other two come "out of the cell" and get multiplied together.   The last 2 doesn't have a partner so it must remain in the jail cell forever, so your answer ends up being 6√2.

So now it is time for you to try one.  Try simplifying : √162,  then check your answer on the next page.