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.
Did you get this?
162 breaks down into 2*9*9.....Since there are two 9's, one dies and one escapes and the 2 doesn't have a partner, so you end up with 9√2 for the answer.
Or if you break down the 9's down so you end up with 2*3*3*3*3, you have two pairs of 3's so for each pair a 3 dies and two 3's come out and get multiplied together and the 2 still doesn't have a partner so it stays inside still giving you the answer of 9√2. Either way you get the same answer.
Now lets talk about rule 2...don't leave a fraction in the square root. Lets look at the example: √¼. What we do is split the square root into two separate square roots, so we get √1/√4. Then we simplify both square roots by taking the √1 =1 and √4 =2 giving us the fraction 1/2. We must make sure the fraction is reduced and since it is in this case, the final answer is 1/2.
Lets try a little more difficult fraction. Suppose we start with the √(3/12). First this time, we can reduce the fraction before we break it down. Therefore we get √(1/4), which gives us the same answer as before : 1/2. But sometimes we can't reduce the fraction to start with. Take the problem : √(50/49). Since we can't break it down before we make two square roots, we go ahead and make two square roots leaving us with √50/√49. Then we take the square root of each ending up with 5√2/7 because we have to break 50 down into 5*5*2 and use the method above to simplify.
Sometimes when we make two separate square roots out of a fraction, the denominator doesn't have a perfect square. According to rule #3, we cannot leave a square root in the denominator, so we must do what we call rationalize the denominator. This means we must multiply the top and bottom of the fraction with the square root of the same number that is in the denominator so that you have two numbers the same so they have a partner and one can escape and one dies. Watch this example....√(1/3) can't be reduced so it must be split up into 2 square roots giving us 1/√3. Since we can't leave a square root in the denominator, we multiply 1 by √3 and √3 by √3 so that the result is √3/3 because when we multiply √3 by √3 we end up with 2 3's so one dies and one escapes, so it is no longer under the radical sign. This leaves us with √3/3 as the final answer because it can't be reduced either. One 3 is under a radical and the other one isn't, so it can't be reduced. To reduce a fraction, either both numbers must be under a square root or both must have escaped and be outside the square root symbol.
Lets try that again. Take the example: √(4/3). First split up the square root into 2 square roots.....√4/√3. The √4 =2 and √3 can not be broken down. So we must multiply the 2 by √3 and the √3 by √3 giving us a final answer of 2√3/3 which can't be reduced.
Now lets have you try one. Take √(16/27) amd simplify it using the 3 rules discussed previously. Then check your answer on the next page to see if you got the correct answer.
Did you get this?
First, split up the square root into two square roots. So now you have √16/√27. Then we break down 16 into 4*4 so √16=4 and 27 breaks down into 3*3*3 which gives us 3√3. This leaves us with 4/3√3. Then we multiply the 4 by √3 and 3√3 by √3 leaving us with 4√3/9 because with 2 3's under the square root, one dies and one comes out and gets multiplied by the 3 already there, leaving us with the 9 and nothing under the square root. Therefore the final answer should have been 4√3/9.