Since Pythagorean theorem proofs requires us to square numbers and find square roots, reviewing square root operations from Algebra is really important. When working with the Pythagorean theorem, it is especially important for you to remember how to simplify square roots and rationalize fractions that have a square root in the denominator.
When you get to the part of Geometry where you have anything that's being squared, especially with Pythagorean Theorem, you have to go back and remember how to simplify square roots. Let me show you 2 ways of simplifying square root of 84, we'll call this method one and method two. So method one is using the prime factors, so you're going to take whatever is, you're taking the square root of and you're going to find the prime factors. So I'm going to start with 2 and 2 goes in the 84, 42 times so I'm just going to keep factoring and I'm going to circle the numbers that are prime. So I'm going to say this is 2 times 21 and 2 is prime 21 is not but I can break it up into 3 times 7. So using method one, what I'm going to do is I'm going to write my prime factors underneath my square root. So I'm going to say this is 2 times 2 times 3 times 7, and then I'm going to say well what's my invisible number here? And since it's a square root, I could write a 2 in there it's in play. So I want to group my terms in squares, so I'm going to say that 2 times 2 is 2 squared, I don't have another 3 to group the 3 with and I don't have another 7 to group to group the 7 with.
And the square root of 2 squared well, if these 2's match then your base is going to come out. So this is kind of a trick, it's not very technical but if something is being squared in a square root your base is going to come out. So that 2 comes out so now we're done with that term and we're left with 3 times 7 which I can say is 21. So this method will always work, it's not very technical because you're not understanding why this base term comes out. The reason it comes out is because taking the square root is the same thing as raising it to the power of one half. So when you have a power to a power remember from Algebra you have to multiply them. So you actually have 2 times one half which is 1, so you're saying that this is 2 to the first times 3 to the one half times 7 to the one half, and one half power is a square root. So that's the reason why this method works.
Method 2. Method 2 says if you're great at factoring in your head. What you can do is you can write the square root of 84 and then you have to ask yourself well what's a square number that will multiply into 84. So by square number I'm talking about 1, 4, 9, 16, 25 all those numbers and I see that I can write this as the square root of 4 times the square root of 21. The reason why this works is because we have the square root being multiplied by a square root, you can, multiply the numbers inside and 4 times 21 is 84. And then you can say well, the square root of 4 is just 2 and the square root of 21 I can't factor anymore and there is no whole number square root. So your answer 2 times the square root of 21. So notice that method 2 uses a little bit less room, it's probably little more time efficient but if you are not confident in your factoring you could do the prime factors and use method 1.