Hi! Welcome to Math Antics. In our last video called “Intro to Exponents”, we learned that exponents (also called Indices) are a special type of math operation. In this video, we’re going to expand on what we know about exponents by learning about their inverse operations, which are called “roots”. That’s kind of a strange name for a math operation, but it will make more sense in just a minute. First, let’s review what we mean by inverse operations. In the video called “What Is Arithmetic”, we learned that inverse operations are pairs of math operations that UNDO each other. For example, you can undo addition by doing subtraction, so addition and subtraction are inverse operations. Likewise, you can undo multiplication by doing division, so multiplication and division are inverse operations. As I mentioned, exponents have inverse operations also. There are operations that can undo them, and those operations are called roots. To see how roots and exponents work together to undo each other, let’s look at the simple exponent 4 to the 2nd power (or four squared). Previously, we learned that this is the same as 4 × 4 which equals 16. Doing this exponent meant going from 4 squared to 16. So now if we want to undo that with a root operation, that involves starting out with 16 and then somehow getting back to the 4 which is being raised to the 2nd power. And do you remember what that part of the original exponent is called? Yep, it’s called the “base”. So doing a root operation is going to give us the base as our answer, and that helps us understand its name a little better. The words “base” and “root” have a similar meaning, especially if you think of a tree. The root is at the base of a tree, and that can help you remember how root operations work. With a root operations, you start with the answer of an exponents and try to figure out what the base of that original exponent is. Okay, but how do we actually do that? How do we use a root operation to go backwards and figure out the base of the original exponent? Well for starters, we need to know about a special math symbol that looks like this. And you guessed it… it’s called the root sign. Whoa Dude! That math symbol looks totally radical dude! It’s… it’s like that division thingy, only way cooler!! Ah yes, that reminds me… the root sign is often referred to as the “radical” sign (and mathematicians used that term even before surfers did) And yes, it does look similar to the division sign so it’s really important not to get them confused. The root (or radical) sign is different from the division sign because, instead of having a curved front, its front shape is like a check mark. The number that you want to take the root of goes under the sign like this. So when you see a number under a root (or radical) sign like this, you know you need to figure out the base of the original exponent. In this case, you need to figure out what number you could multiply together a certain number of times to get 16. Ah - but there’s the catch! How many times? The answer we get from taking the root will depend on how many times that number would be multiplied together. But that would depend on the original exponent. So how do we know what that number is? Simple… the root symbol tells us. The root symbol actually includes the original exponent in it. What? You don’t see it? Oh… That’s because I didn’t draw it yet. And later in this video, you’ll understand why. So let’s put a little 2 right here above the check mark part of the root symbol. And that 2 tells us that we need to figure out what number (or base) could be multiplied together 2 times in order to get 16. And, if you remember your multiplication table (or if you just look at our original example here) you’ll know that the answer to that is 4. Now do you see how the root operation is the inverse of the exponent operation? When doing the exponent, we asked, “What do we get if we multiply 4 together 2 times?”, and the answer was 16. But when we did the root operation, we asked, “What number could we multiply together 2 times to get 16?” and the answer was 4. Great! Now that you understand how exponents and roots are related, we’re going to look closer at how root operations work. To do that, we’re gonna change our root problem slightly. Let’s change the little 2 into a little 4. The first root was asking us to figure out what number we could multiply together 2 times to get 16. But this new root is asking us to figure out what number we could multiply together 4 times to get 16. That’s a bit trickier, huh? Can you think of a number like that? Yep, the answer is 2, because if you multiplied four ‘2’s together (2 × 2 × 2 × 2) you get 16. So the 2nd root of 16 is 4, but the 4th root of 16 is 2. Both those roots were pretty easy to figure out, right? But unfortunately, figuring out roots in math can be much harder. For example, what if we had this problem instead, “root 3 of 16” That means we need to figure out what number we could multiply together 3 times to get 16. Can you think of a number like that? [No] I can’t either! And unfortunately, it’s not easy to calculate what that number would be. Remember, even hough this look a little but like the division symbol, this is NOT just division! You can’t just divide 16 by 3 to get the answer. Roots are NOT the same as long division. So how DO we calculate a root like this? Well, there are special algorithms that you can use to calculate just about any root, but they’re kinda complicated, so we’ll save those for a future video. Instead, I’m gonna use a special root function on my calculator to get the answer. And on my calculator the button for that root function looks like this. To use it, I first enter the number that I want to take the root of, which is 16. Next, I hit the root function button, and then I enter 3 so it knows that I want the 3rd root of 16. Last, I hit the equals sign and voila… the answer is 2.519842… and the decimal digits just keep on going forever. Wow! See what I mean about roots being hard to figure out? That is a really complicated decimal number and you may even wonder if it’s the right answer. Well, let’s check… Based on what we know about exponents and roots, if we multiply this decimal number together 3 times, we should get 16, right? But to make it easier to check, let’s just round the number off to 2 decimal places. Let’s make it 2.52. If we multiply 2.52 together 3 times, (in other words if we take 2.52 to the 3rd power) we’ll get: 16.003. Well… that’s almost right. It’s really close to 16, isn’t it? The reason it’s not exactly 16 is that we rounded the number off which made it less accurate. But the more decimal digits we use, the closer we’ll get to 16. In math, the vast majority of roots are complicated number like this. And they’re hard to figure out unless you use a calculator or a special algorithm. That’s the bad news. But the good news is that most of the time, the roots you’ll be asked to do in your homework or on tests are the easy ones; ...the ones that have nice whole number answers. And usually, you’ll only be asked to find 2nd or 3rd roots of numbers. Do you remember in the last video, we learned that 2 and 3 are the most common exponents. …so common in fact, that they even had special names. Raising a number to the 2nd power was called “squaring” it, and raising a number to the 3rd power was called “cubing” it. Well, it’s the same with roots. Since the roots 2 and 3 are the most common, they get special names also. The 2nd root is called the “square root” and the 3rd root is called the “cube root”. In fact, the “square root” is SO common that it’s basically the default root and its symbol even gets special treatment. Do you remember that when I first showed you the root symbol, I left out the index number that tells you what root to find? Well, whenever that number is left out, you can just assume that it’s 2. In other words, the root symbol, with no index number, is always the square root. So if you want someone to find a different root (like cubed or 4th or 5th) then you need to include that number so they know which root to find. And even though square roots are the most common, they’re not always easy to find. Most are still going to be big long decimal numbers, except for the “perfect squares”. It’s easier to find the square roots of the perfect squares because their answers can be found using the multiplication table. On the multiplication table, have you ever noticed that all the answers to problems where the same number is being multiplied together are on the diagonal of the table. In other words, 2x2=4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25, 6 × 6 = 36, and so on. Well, those numbers are called the perfect squares because they’re the answers you get when you square a whole number. And that means if you take the square root of a number along that diagonal, you get a nice whole number as your answer. The square root of 4 is 2. The square root of 9 is 3. The square root of 16 is 4. The square root of 25 is 5, and so on. See what I mean? Those roots are really common and they’re also easy to figure out if you know your multiplication facts. So if you’re new to exponents and roots, learning the perfect squares is the place to start. Once you understand how those exponents and roots work, you’ll be ready to figure our tougher problems. Alright… so now you know how exponents are related to roots. They’re inverse operations and they undo one another. And you also know that, just like 2 and 3 are the most common exponents, the square root and the cube root are the most common roots. You also know that finding roots is usually not very easy. That’s important to know so you don’t get discouraged if you feel like it’s hard to figure out what a certain root is. You’re not alone! We think it’s hard too and would normally just use a calculator to find them. The good news is that some roots are easy to find, like the perfect squares, so be sure to focus on learning them first. And remember, to get good at math you need to actually practice what you learn from watching videos, so be sure to do some exercise problem. And as always, thanks for watching Math Antics and I’ll see ya next time. Learn more at www.mathantics.com