hey everyone welcome back to mashup math Anthony here and on this lesson we get to visually explore that relationship between square roots and perfect squares so we really get to visually explore that question what exactly is a square root let's go ahead and take a [Music] look hey everyone welcome back to this introductory lesson on Square roots and perfect squares so we are going to start off our discussion by looking at a rectangle and this rectangle has a width of four units and a length of eight units now we're going to Envision this in terms of area so that width of four think of that as four equal size units that will split it up along the width and then for the length we have eight equal size units which we will split it up along the length and now we have these squares that make up the inside the area of that figure length time width in this case 4 * 8 which we know is 32 so the area of this rectangle is 32 square units so now let's look at what would happen if we cut this rectangle in half so now instead of it being a 4x8 rectangle it's a 4x4 which makes it a square since all of the sides have the same length the area of this Square 4 * 4 we know is equal to 16 s units and this relationship allows us to say that 16 is a perfect square and also that the square root of 16 is 4 since 4^2 = 16 and we can think of this relationship every time we think about perfect squares now for example if we took this 4x4 square and cut each unit in half to make it 8 by 8 now if we counted up all those squares to find the area or just multiply 8 by 8 8^ SAR we would get a result of 64 square units so 64 is also a perfect square and the square otk of 64 is 8 because 8 SAR will equal 64 so now we can say that a perfect square is a number that can be made by multiplying a whole number by itself by squaring any whole number cool so now we can revisit the two perfect squares from earlier and that was the 4X 4 square and the 8X 8 square now 4 * 4 if we think of this in terms of area is 16 which is equal to 4^ 2 and 8 * 8 is equal to 64 which we know is equal to 8 SAR notice again that four and 8 are both whole numbers now if we take the square root of 16 our result is four and if we take the square root of 64 our result is 8 so the most important thing to take away from this lesson is that relationship between a perfect square and its square root and now we can explore that relationship a little bit more let's take a look at an example of a perfect square of 25 and a non-perfect square 30 now 25 is a perfect square because 25 is equal to 5^ SAR and the square root of 25 is equal to 5 which is a whole number so by definition everything works out a non-perfect square 30 if I take the square root of 30 I get approximately 5.48 which of course is not a whole number so there is no whole number that we can square to get to 30 the way that we can square five to get to 25 now 25 and 30 are pretty close to each other in value and if we think about these two numbers in terms of area again we know that 25 is a perfect square that has Dimensions 5 by five and if we think about that area in comparison to that 30 if we slide it over we see that it kind of fits but there is some room left over that orange space which doesn't quite hold one full Square it only holds about half of a Square which is where that 48 that decimal comes from and again this is because there is no whole number that we can square to get to 30 we'd have to have a decimal value in order to approximate a square root for a number like 30 so now we're going to go ahead and Visually explore some of the most common perfect squares and their relationship to their square root so one 2ar is 1 * 1 which just equals 1 2^ 2 is 2 * 2 which = 4 3 2 is 3 * 3 which = 9 4 2 again 4 * 4 which equals 16 again these are all perfect squares 5 squar is 25 6 SAR 6 * 6 = 36 7 SAR is 7 * 7 which = 49 8 2ar is 8 * 8 which equal 64 we actually looked at that one earlier and 9 s 9 * 9 is equal to 81 and the square root for each one of these perfect squares is the value that you had to square to get it in the first place so for example the square < TK of 4 is equal to 2 the square < TK of 9 is equal to 3 the square root of 16 is equal to 4 and so on and so forth and this is why raising a number to the second power or the power of two is referred to as squaring the number so those are the basic concepts involved with per perfect squares and square roots so keep that in mind as you continue to build upon that understanding and apply it to algebra and more advanced levels of mathematics and we'll catch you guys next time all right so that's it for that lesson hope you found it helpful and if you did please click that link below and subscribe to our YouTube channel we add new lessons every week we don't want you to miss out and also if you have any questions or concerns feel free to comment down in the comment section below we respond to every single comment I promise you will respond even the meme ones okay but let's just try to keep it nice those ones are always a lot more fun to read and uh we'll catch you guys next time see you