>> We want to simplify the given square roots.
We know from the previous examples that if A squared is equal to B, then A is the square root
of B, but notice in this case there's not a number times itself that's equal to 50,
but a square root is considered simplified if it contains perfect square factors.
So we do want to simply this by identifying the perfect square factors,
and the best way to do this is to look at the prime factorization of the of the radicand
with a number underneath the square root, and one last thing to notice,
these are the principal square roots, so we're only concerned about the positive square roots.
So for the prime factorization of 50, we can write 50 as 2 times 25, and 25 is 5 times 5.
So the prime factorization of 50 is 2 times 5 times 5.
Well, any time we have two equal factors, we have a perfect square factor.
So here's our perfect square factor, so this can simplify, and the square root
of 5 times 5 is equal to one factor of 5.
So we have one factor of 5 times a remaining square root, which is the square root of 2,
so again, whenever we can identify two equal factors,
that's a perfect square factor, and it will simplify.
Now let's try the same for the square root of 675.
So we'll start by writing out the prime factorization of 675.
Well, it ends in a 5, so we know it's divisible by 5.
675 is 5 times 135.
5 is prime.
135 ends in 5, so it's divisible by 5.
This would be 5 times 27.
5 is prime.
27 would be 3 times 9, where 3 is prime and 9 is 3 times 3.
So we have three factors of 3 and two factors of 5 for the prime factorization of 675.
So we can write this as 3 times 3 times 3 times 5 times 5, and now we'll group paris of factors.
So here's a perfect square and here's a perfect square.
So the circled part is going to simplify,
and the factors not circled will stay underneath the square root.
So notice how the square root of 3 times 3 is 3.
The square root of 5 times 5 is 5.
So these will be outside the square root, because they simplify,
and then we're left with the square root of 3.
So this would be 15, square root 3.