all right now simplifying radicals is a is a is an important key to a lot of the things we study in chapter 8 on right triangles so I'm going to give you a few examples of simplifying radicals into a simpler radical form that doesn't mean a decimal that just means a simpler form that still has a radical in it um now our goal for simplifying radicals has to do with the square roots of perfect squares like the square root of four is two the square root of n is three the square root of 25 is 5 square of 100 is 10 you get the idea so when we have the square root of a perfect square um it gets pretty easy for us to um to simplify it so we're looking for perfect squares as we are simplifying so I will take 12 and make a factor tree with it 12 is is let's see here 3 * 4 um and 4 is 2 * 2 now whichever factors you choose you should still get three 2 and two so the square of 12 is equal to theare < TK of 2 * 2 * 3 now since I have a pair here the two and the two it gives me a perfect square so I'm going to write this as 4 * 3 now if I have the square of two things being multiplied I can separate those into two separate square roots okay and the square of four I know to be two and the square of three is still the square Ro of three and so this becomes um simplifies to be two sare of three okay now in practice I don't usually go through all these steps because it takes too long in practice I look I do my factor tree I find my pair now when I have a pair that means I have a perfect square and for that pair I take one of those numbers and write them outside of my square root and then whatever's left without a pair without a perfect square um stays inside so by two my pair of twos I have a two here and the three I have a square Ro of three so we just jump from here to here me give you example of that um the square of 90 now 90 I'll just pick three and 30 um 30 is three and 10 10 is 2 and five so I have a pair of three and three that means I have a perfect square so I write this as three from my pair of 3 is square root of what's left 2 * 5 and 2 * 5 is 10 and so 3un 10 is a simplified um form of the square Ro of 90 okay so this a little more complicated 3 * 2 is 6 and the square of 3 * the sare of 3 is a square Ro of 9 okay and the square of 9 I know is three so 6 * 3 is equal to 8 18 okay now this next one's a little more complicated um 6 * < tk2 is 6 < tk2 and the square of 2 * the square of of two is the square < TK of 4 okay I'll rewrite that down here 6 root2 over theare of 4 now the sare of 4 is equal to 2 oh now 6 over2 simplifies to be 3 over 1 and I got 3 < tk2 which is a simpler form 8.1 is about the Pythagorean theorem okay if a right Tri if a triangle is a right triangle then um the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse now to me that doesn't make as much sense as just thinking that if it's a right triangle then A2 + B2 equal c^2 where C is the hypotenuse okay it only works for right triangles now when a B and C are all whole numbers we have what's called a Pythagorean triple for example 3 4 and 5 3^2 + 4^2 = 5^2 they're all whole numbers it's a Pythagorean triple 5^2 + 122 = 132 it's a Pythagorean triple so when this happens I'll refer to it as a pyan triple okay so the pyan theorem is one of the most proved theorems I think in all of mathematics there's hundreds of proofs uh that are slightly different that show different reasons why A2 + B2 equals c^2 here's one example um here's a triangle of the height is a the length is b and the hypotenuse is c um now if you look down here this length down here is still C and this length is C and the shape of it is a square so the length of this down here well sorry the area of this is c squared okay so this whole these pieces together make up C squ keep that in mind all right now if I begin to move these lengths um the area is not going to change so this big area is still equal to c^2 I move this one as well still equal to c s but an interesting thing happens this length is a and this length here is also a and so this the area of this square is a s and the length of this sorry this length is b and this length up here is B and so this square right here is b s so you can see that a s+ b^2 is equal to c^2 here's an example find the length of the hypotenuse I'll call the hypotenuse X and since it's a right triangle I know that A2 + B2 is equal to c^2 now C is always the hypotenuse in this case the hypotenuse is X so C is X um A and B can be either 20 or 21 it doesn't matter so 202 + 212 is x^2 now 21 is 400 sorry 20 squ is 400 21 squar is um 441 which is equal to x^2 okay adding these together 841 is equal to X squ so let's see here how do you undo a squared you do a square root so X is the square root of 841 which equals 29 okay it's a perfect whole number and so 20 21 and 29 are a Pythagorean triple okay here's an example where we want to find one of the legs again a 2 + b^2 = c^2 works because it's a right triangle now C is always the hypotenuse so in this case C is 20 let's see here um I'll say x^2 + 8 2 is = to 20 2 x^2 + 6 2 is 64 202 is 400 now to solve for x^2 I'm going to subtract 64 from both sides of my equation x^2 is = to 336 okay to simplify X squar I get um X is equal to the square < TK of 336 now to simplify 33 36 I'll need to draw a factor tree 336 is let's see here 3 * 112 112 is 2 * um 56 56 is um 7 * 8 8 is four and two and two is two and two all right so I have some pairs I have this two and two and this two and two and so this simplifies to be 2 * 2 the square of what's left 3 * 7 and so I get 4 < tk21 okay the converse of the Pythagorean theorem again the Pythagorean theorem is if A2 plus b^2 I'm sorry the Pythagorean theorem is if it's a right triangle then A2 + B2 equal c^2 the converse where you take those at if and the then and you switch them so the converse is if a s plus b equal c^2 then triangle ABC must be a right triangle so question a triangle has side lengths 10 24 and 26 is the triangle a right triangle well we can test to see if a^2 + b^2 is equal to c^2 and now of the three the longest one's 26 so that must be C so I'll say 10^2 + 242 and now I don't really know so I don't want to put equals I'm put equals with question mark above it is equal to 26 SAR okay so 10 squ is 100 24 SAR I don't know left top of my head 24 squared is 576 is equal to question mark 676 so 100 + 676 is 676 and that's equal to 676 and so this must be a right triangle because it's equal okay another example a triangle has side length 6 9 and 12 is the triangle a right triangle again I'll test a^2 plus b^2 to see if it's equal to c^2 now 12 is the longest that must be c 6^ 2 + 9 2 to C is that equal to um 12 2 6 2 is 36 9 squar is 81 and 12 squar is 144 36 + 81 is 117 but that's not equal to 144 okay so this is not a right triangle now one interesting idea we can think about is if it's not a right triangle is it acute or is it obtuse and the way I think about that is I think is C too big or is it too small in this case it's too big if it's too big it is an obtuse triangle okay so it's not a right triangle it happens to be an obtuse triangle again if C is too small or Too Short then it will be an acute triangle if C is too big or too long it's an OB