let's look at uh perfect squares and square roots and their relations I'm going to do is I'm going to draw a square and I'm going to make this a 4x4 okay so I've got a square here its side here is four cm long and this side is 4 cm long so see it's a square and then we can see that the area inside of this square is 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 cm squar or of course we could have just gone length time width 4 * 4 and got our area of 16 cm squar so perfect squar are numbers that we can multiply together that have the same length and the same width or another way to write that is we can say a perfect square is a number that can be expressed as the multiple of two factors that are the same so they would just be the length and the width of the sides of squares we can do another one let's get blue back again why don't we we do a 5x five 1 2 3 4 5 1 2 3 4 5 so I've created another perfect square this one is a 5x5 and when I multiply these two sides together I generate that number 25 so 25 is another perfect square and we could make a list of of the first five 1 * 1 would be the smaller one smallest one which is one we could do a 2X 2 square which would generate an area of four units 3 * 3 would be the next one 9 4 * 4 you already did that one 16 and 5 * 5 would be 25 so these numbers here in this column are the first five perfect squares they're numbers that we would generate when we start with a factor and multiply it by itself and of course we could do more we could generate more perfect squares 6 * 6 36 is the next one 7 * 7 49 so those are our first seven perfect squares now related to that is a square root and a square root symbol is usually denoted by this symbol right here and so if in for instance if I was asked to find what is the square < TK of 16 what that is meaning is what is the number that when I multiply it by itself gives me 16 well we just have to look at our little table here to realize that the number is going to be four so we can say that the Square < TK of 16 is 4 another way of looking at this then is the square < TK of 16 if 16 is the area that we want what would be the lengths of the sides the lengths of the sides would need to be four why because 4 * 4 is 16 so these are our perfect squares the numbers in this column then would represent the square root of these numbers here let's say we would want to find the square root of 144 there are many ways that we could determine what the square root is one would be to take a box and begin to count 144 squares this would take some time to do do you'd have to count them all um but if we did this one two three four five if we did this here I believe what I've done here is I have created a square that has 144 squares in it if you count all those you will see there's 144 squares here the length of this side is 12 1 2 3 4 5 6 7 8 9 10 11 12 and this is 1 2 3 4 5 6 7 8 9 10 11 12 so this would be one way to determine the Square t of 144 we would see that the length of a side would be 12 U not maybe the easiest way of doing things um but nevertheless if you had some Square paper and you felt like counting 144 squares uh you would see that the length of each of your sides would need to be 12 another way that you could figure out the Square < t of 144 is we could use our our table here of of perfect squares so I think we had up to 7 * 7 which was 49 before so that's too small 8 and 8 64 9 * 9 is 81 we're getting closer to 144 we got to keep going 10 * 10 is 100 11 * 11 is1 and then 12 * 12 we'd see is 144 so we would know that the square of 144 must be 12 because 12 * 12 is 144 that would be a second way a third way would be to do a prime factorization tree so let's do that let's take 144 and we know we can divide that by two 72 we can divide by two 36 we can divide by two 18 we can divide by two whoops 18 2 * 9 and 9 we can divide by 3 so we can write 144 as 2 * 2 * 2 * 2 * 3 * 3 and you'd want to this is a long one so you'd want to double check that you did indeed list them all here by multiplying them out and making sure you get 144 uh we're good with that so let's move on now notice when we've done this I've got a pair of twos I've got another pair of twos and I've got a pair of Threes so since we're finding the square root this is like saying we've got a 2X two square here and another 2x 2 square and then a 3X3 Square so if I want to find the square root of 144 I can just take half of my factors so the 2x two is is has a side length of two this 2x 2 square would have a side length of two and the 3X3 Square would have a side length of 3 so theun of 144 would be 2 * 2 * 3 2 * 2 is 4 4 * 3 is 12 so I can find the square roots of numbers by doing a factor tree and as long as there's pairs of each of the terms I can take take what number is in each pair multiply those together and I get the square root and of course you also probably on your calculator have a square root function built in and you could go on your calculator the square of 144 and it would tell you that the square root is 12 so there's four different ways in which case we can find the square roots of numbers