 Hello, this is a video about simplifying radicals. First, we need to understand what a radical expression is. It is read as the nth root of a, where in that little number outside what we call the radical sign is the index, the quantity under the radical is called the radicand, and then that sign is actually called a radical sign. When there is no index given the radicals understood to be a square root, square roots always have an index of 2. The reverse operation of squaring a number is finding its square root. It's important to note that when we take the square root of a quantity, it's always going to be a positive result that's for square roots. The only time the result will be negative will be when there is actually a negative sign outside of the square root. In case you're wondering and your calculator will verify this, the square root of 0 is equal to 0, and other important thing is that if you ever have a negative sign under an even index radical, such as a square root, the answer is not defined as a real number, it's not real. But if you have a negative sign underneath an odd index radical, such as a cube root with an index tree, well then you're allowed to bring that negative sign out front of the radical. So I have several square roots in their perfect squares and in perfect cubes in their cube roots displayed on the screen. It is recommended that you memorize the perfect squares in their square roots and at least half of the perfect cubes in their cube roots. For example, if you see the square root of 16, that's saying what number when multiplied with itself twice because you're doing square roots is 16. Well that would be 4. Cube roots, cube root of 125. Since we're dealing with cube roots, what number when multiplied with itself 3 times gives you 125? Well that would be 5. Your calculator would also evaluate all these perfect squares and perfect cubes for you as well, but it saves time to memorize them. So now, example one part A, the square root of 36, it's a square root. What times itself gives you 36? 6. Part B, the negative goes along for the right and you take the square root of 4, which is 2. Cube roots, the cube root of 8 means what number when multiplied with itself 3 times gives you 8? Well that's 2. And then 64, similarly what multiplies with itself 3 times to give you 64, and that is 4. Alright so now we're going to actually go in and talk about simplifying variables under radicals. It's actually as simple as taking the power on the variable and dividing it by the index. So for part A, the index is 2. How many times does 2 go into the power 4? Twice, with the remainder of 0. What that means is that you pull out a z squared and then there should be no z remaining under the radical. Part B, the cube root of 27 is 3 and your job using an index of 3. How many times does 3 go into 6? Twice, with the remainder of 0. So there should be no y's that remain under the radical. So the answer will be 3y to the second. Part C, square root, the index is a 2. Square root of 16 is 4. How many times does 2 go into 8? 4 with the remainder of 0. No x's will remain under the radical. Next, part D, you see that the index is odd. That means the negative sign can actually be brought out front of the radical. Then we'll continue simplifying the inside. So I'll have negative 2 times whatever the cube root of 8x to the 9th becomes. Well, cube root of 8 is 2. How many times does 3 go into the power 9? Well, it should be 3 with the remainder of 0. No x's remain under the radical. So I have 2x cubed is what the radical simplifies into. Multiply all that by negative 2, giving me negative 4x cubed. Unfortunately, we're going to run into times when the number under the radical is not a perfect square. When the number under the cube root is not a perfect cube. And times when the power on the variable is not perfectly divisible by the index. Well, we need two properties to handle this sort of situation. And the first property says that if you have things multiplied together under a radical, you can actually write them as being under a radical separately. Multiply situation will split under a radical. The vision also splits under radicals. So example 4 part A, you say the square root of 24. It's not a perfect square. You can even check your table just to be sure. Since 24 is not a perfect square, we have to think. What perfect square is 24 divisible by? Well, think about it. 24 divided by 16 doesn't happen evenly. 24 divided by 9 doesn't happen evenly. 24 divisible by 4. Yes, it's 6. So what you do is you write 24 as 4 times 6. Since radicals split under multiplication, split splits under radicals, you can write this as square root of 4 times square root of 6. Square root of 4 is 2, square root of 6, no can do. The radical is now simplified. Now, same thing here. Square root of 75 with a 3 out front. 75 is not a perfect square, but it is divisible by 25, which is. 75 can be rewritten as 25 times 3. So this is 3 times the square root of 25 times the square root of 3. This is 3, square root of 25 is 5 times the square root of 3. In the end, this becomes 15 square root of 3. So it's all about trying to find out what perfect square divides evenly into the number under the radical. Part C, you have division. Square root of a fraction means you take the square root of the top over the square root of the bottom. This means I will have square root of 25x squared is 5x. Square root of 36 is 6. More. You see, 18x to the fourth, y to the fifth, I'm taking a square root. So I have the square root of 18 is not a perfect square, but it is divisible by 9, who is. So I have 9 times 2, and you have your x to the fourth, y to the fifth. So this means square root of 9 is 3. We write everything else that's under the radical. And my last step here is to take my index, which is 2, and then divide every power on every variable by 2. So let's look at for x. How many times does 2 go into 4? Twice. With the remainder of 0. So there should be no x's which remain under the radical. So that means I have 3x squared. Let's look at for y. How many times does 2 go into 5? It goes in twice with the remainder of 1. So I can pull out y squared. And then inside the radical, I still have my 2, and then I have my y to the first. So the remainder is the power that remains on the variable inside the radical. Let's go into cube root mode. 32 is not a perfect cube, but it is divisible by a perfect cube. It's not divisible by 27. It's divisible by 8. 32 is 8 times 4. So you have cube root of 8 times cube root of 4. Cube root of 8 is 2. Cube root of 4 you just can't do. Remember you're in cube root mode, not square root mode. Last but not least in part f. You're in cube root mode. 16 can be broken up into 8 times 2. Cube root of 8 is 2. Then rewrite everything else that's under the radical. My index is 3. So how many times does 3 go into 9? And this is for a. Then for b. How many times does 3 go into 11? 3 goes into 9 3 times with the remainder of 0. Then you can pull out 2, because that's already there. A cubed. For b, 3 goes into 11 3 times with the remainder of 2. So I can pull out b cubed. And the remaining underneath will be b squared. And there's also that 2 that's still under the radical. So this is all fully simplified and never confuse the index with one of the powers on the variables. So be careful with that. And one thing to always check your answer is, if you ever have an exponent under the radical greater than or equal to the index, your answer is not correct or not fully simplified. Notice that the power on b is 2. The index is 3. Notice the power on y is 1. The index is 2. It's not written there because it's a square root. So it is fully simplified. So thank you for watching.