Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. We're going to look at radicals square roots. The number C is the square root of A. If C ^2 equal A the principal square root. The principal square root of a non negative number is the non negative square root. So back here if we said C ^2 equal a, we could think about taking the square root of each side. And when we take the square root we think about having both positive and negative. So we would get C equaling the positive and the negative square root of A. For the principal square root we're just looking at the positive. And the reason this is true is we could think about -1 ^2 is 1, but also 1 ^2 is 1. The radical sign is just the symbol with the radical right here. Frequently we have a number right here and that number is referred to as the index. The expression under the radical sign is called the radicand sqrt. A ^2 is just the absolute value of A and the reason for this is underneath an even index, IE and even index. Here the inside has to be a positive, so even index the inside has to be positive or zero. If it's an odd index, the inside can be anything and the reason is odd index inside can be anything, and the reason is that if it's a negative on the inside and it's odd, we can have an odd quantity giving us a negative. IE 3 negatives multiplied together would give us a negative cube root. The number C is a cube root of A. If C ^3 equal a, the notation is the cube root of a, the NTH root. So the NTH root of eight in the N N is even and a is positive. Then the NTH root of 8 and the N equal a. And the reason is if I have the second root, actually let's do the 4th root of say 3 to the 4th, 3 to the 4th is a positive number. It's 81. And then the 4th root of 81 is going to be the three. If a is a negative, though, let's say we have the 4th root of a -3 to the 4th. What's going to happen is that -3 to the 4th is going to end up becoming a positive number. And then when we take the 4th root of it, it's the fourth root of a positive number. So it would be the equivalent of the absolute value of that negative number, or just +3. If N is odd, then it can be anything. Say I have the cube root of -1 ^3 while -1 ^3 is -1 and the cube root of -1 is the -1 and it's called the index. Let's look at some examples. Sqrt 49 we could think of as 7 * 7, so sqrt 7 ^2. The square root and the square are going to cancel, leaving us just 7. This next one we have a negative outside 81 is just 9 * 9 one 44 is 12 * 12. So if we have sqrt 9 ^2 / 12 ^2, the square root and the squares are going to cancel. So we're going to get sqrt 9 twelfths, but nine twelfths is actually going to reduce to 3/4. We could have actually started and done this one a different way. Also, we could have reduced inside 1st 81 is divisible by 944 is also divisible by 9. So if we had seen that 81 was divisible by 9 and 144, nine goes into 14 one time with five leftover and nine goes into 54 six times. So now sqrt 9 is 3, sqrt 16 is 4. Lots and lots of different ways to do things. Find index and radican. So the index is just this number 3 here. So the index is 3. The radican was just what was underneath the root sign. So a / a ^2 -, b sqrt 25 T squared. So we have 25 is 5 ^2, and then we have t ^2 and the square root and the square cancel. We know that 5 is positive, but we don't know if T is positive or negative. So the way we're going to write this answer is we're going to use the absolute value, and we're going to say five times the absolute value of T because if T had been a negative number when I squared it, it would have turned positive #5 it's not a monomial. It's not a single term underneath. So we actually have to factor it to start X -, 2 X -2. So then we get sqrt X - 2 quantity squared, and the square root in the square are going to cancel. But once again, we don't know whether this X - 2 is positive or negative. So we're going to put it in absolute values. Now this next one, it's a fifth root. So we actually need to figure out how many times what number times itself five times. So I'm going to start with 32 into a factor tree. 2 * 1616 is 4 * 4, so 2 * 2 * 2 * 2 * 2. So 16 * 2 was the 32 or 2 to the 5th this 243. If we're not aware, divisibility test by three says add up all the digits and if the sum of the digits is divisible by three, then the original number has to be. So 2 + 4 + 3 is 9. So I know that 243 is divisible by three. Three goes into 20, four 8 * 3 goes into that three. One time 81 is 9 * 9, so we get 3 to the fifth being the 243. So when I take the 5th root of a negative, first of all that's a negative, and the 5th root of 2 to the 5th is 2, and the 5th root of 3 to the 5th is 3, so negative 2/3. Here's a few for you to try, and then let's look at these next examples. The square root and the square are going to cancel, and we're going to have to have that absolute value because we didn't know if 5 + b started out as positive or negative. Now the shortcut for this, this isn't understood to here. Whenever there's not a number, it's understood to be a square root. So think about two goes into the exponent of 14 how many times? And the answer 7. But because it was an even index, we need to have absolute values around it. So the shortcut method is the index goes into the exponent how many times? And if the index started out as even, we have to put absolute value G of X = sqrt X + 8 inside the square root must be positive to 0 if the index is even. We've talked about that already. So if we wanted to think about the domain here, we know that the domain has to have the X + 8 or the inside greater than or equal to 0. So X would be greater than or equal to -8. We want to make sure we remember interval notation. So we're going to say -8 out to Infinity. If we thought about drawing this on a number line, we'd have a solid dot and we'd have it shaded off to the right. Now this next one wants to know the domain again. And because the index is odd, the domain is all real numbers. So in this case, we literally just write negative Infinity to Infinity and we're done. And that's all because that index was odd. If the index is even, the inside has to be greater than or equal to 0. If the index is odd, it's just all reals and we're done and we move on. So this next one, we want to figure out this the domain. So seven X -, 5 has to be greater than or equal to 07 X greater than or equal to 5 X greater than or equal to 5 sevenths. Putting it in interval notation, we have 5 sevenths out to Infinity. And then there's a few for you to try. These are just simplified. These are not domains. Thank you and have a wonderful day.