Hello wonderful mathematics people, this is Anna Cox from Kellogg Community College. Some definitions of radical expressions. Square root. The number C is a sqrt a if C ^2 equal a principal square root. The principal square root of a non negative number is its non negative square root. Radical sign. It's the symbol with the radical. The expression under the radical sign is the radic hand. Sqrt a ^2 equals the absolute value of a because the inside of a square root always has to be positive. So if we had sqrt 9, that would be 3 because sqrt 3 ^2, the square root and the square root cancel. It's also true that sqrt -3 ^2 would be the absolute value of -3, or in this case 3, because the square root and the square cancel. But we need the absolute value because the inside of the square root had to be a positive. The number C is a cube root if a of a. If C ^3 equal a, The cube root of A is the notation, the NTH root the NTH root of A to the north. If N is even and a is positive, then a solution exists. If N is even and a is negative, then no real number solution exists. Remember that if we have the NTH root of negative A to the north, that negative really means -1 * A to the north, and the NTH root, if it's even of a negative number, is going to be an imaginary. We're going to study that later. If N is odd, then a real number solution exists. N is called the index. So some examples, sqrt 49 are 7 and -7 because if we have sqrt 49 we could think of that as sqrt 7 ^2 which is 7. But it could also be thought of as sqrt -7 ^2 or the absolute value of -7. So 7 and -7 are both possible square roots. We usually only talk about the principal square root, which is 7. We have sqrt 81 / 144 and the negative of that, the negative out in front. We're going to just do the top and the bottom separately. So sqrt 81 is 9, sqrt 144 is 12, so -3 fourths. We can also reduce on the inside to start. These are each divisible by 9, so we could have had sqrt 9 / 16, so negative sqrt 9, which is 3 / sqrt 16 which is 4. So more than one way to do that. Same answer either way. The next one was asking what the index is. The index is the number or the N when we're having the NTH root. So in this case the index is 3. The radican is what's ever inside. Or a / a ^2 -, b #40 sqrt 25 T squared sqrt 25 is 5 sqrt t ^2. We don't know what that T is, so we have to use absolute value whenever we have a variable and an even index. If there's no number, it's understood to be a square root or A2 for its index. We have to use absolute value for variables 46. Sqrt X ^2 - 4 X +4 is the quantity X - 2 ^2, and the square root in the square are going to cancel, leaving us the absolute value of X - 253, the 5th root of -32 / 243. Well, the fifth root of a -32 negative 2 * -2 * -2 * -2 * -2 is -32 and the 5th root of 243 is 3, so the 5th root of the -32 / 243 is really just negative 2/3. Simplified directions change. We want no negative quantities to even powers, so there are no negative quantities. So none of the variables are negatives. So a square root and a square are just going to cancel. If it didn't say there were no negative quantities, we would put it within an absolute value, the square root of A to the 14th. Because there are no negative quantities, it's just A to the 7th. If it didn't say there are no negative quantities, it'd be the absolute value of A to the 7th 94. We want the domain and we know that the inside of a square root has to be positive or zero. So we're going to take that inside piece and make it greater than or equal to 0 and solve take the 8 to the other side X greater than or equal to -8 98. Because it's an odd index. The inside can be positive, negative, or zero, which makes the domain all real numbers. If we thought about the cube root of 8, that'd be two. The cube root of 0 is 0. The cube root of -8 is -2. So odd indices we can have positive, negative and 0 one O 2. We want the inside to be positive or zero because this index is even, it's a four. So seven X - 5 greater than or equal to 0. Add the five divide by 7, we get X greater than or equal to 5 sevenths or 5 sevenths to Infinity. Thank you and have a wonderful day.