Full transcript
0:06Hi! Welcome to Math Antics.
0:08In our last video called “Intro to Exponents”, we learned that exponents (also called Indices) are a special type of math operation.
0:16In this video, we’re going to expand on what we know about exponents
0:20by learning about their inverse operations, which are called “roots”.
0:24That’s kind of a strange name for a math operation, but it will make more sense in just a minute.
0:29First, let’s review what we mean by inverse operations.
0:33In the video called “What Is Arithmetic”, we learned that inverse operations are pairs of math operations that UNDO each other.
0:41For example, you can undo addition by doing subtraction, so addition and subtraction are inverse operations.
0:49Likewise, you can undo multiplication by doing division, so multiplication and division are inverse operations.
0:57As I mentioned, exponents have inverse operations also.
1:00There are operations that can undo them, and those operations are called roots.
1:06To see how roots and exponents work together to undo each other,
1:09let’s look at the simple exponent 4 to the 2nd power (or four squared).
1:15Previously, we learned that this is the same as 4 × 4 which equals 16.
1:20Doing this exponent meant going from 4 squared to 16.
1:25So now if we want to undo that with a root operation,
1:28that involves starting out with 16 and then somehow getting back to the 4 which is being raised to the 2nd power.
1:35And do you remember what that part of the original exponent is called?
1:39Yep, it’s called the “base”.
1:41So doing a root operation is going to give us the base as our answer, and that helps us understand its name a little better.
1:49The words “base” and “root” have a similar meaning, especially if you think of a tree.
1:54The root is at the base of a tree, and that can help you remember how root operations work.
2:00With a root operations, you start with the answer of an exponents and try to figure out what the base of that original exponent is.
2:08Okay, but how do we actually do that?
2:11How do we use a root operation to go backwards and figure out the base of the original exponent?
2:17Well for starters, we need to know about a special math symbol that looks like this.
2:21And you guessed it… it’s called the root sign.
2:24Whoa Dude!
2:25That math symbol looks totally radical dude!
2:29It’s… it’s like that division thingy, only way cooler!!
2:32Ah yes, that reminds me… the root sign is often referred to as the “radical” sign
2:38(and mathematicians used that term even before surfers did)
2:42And yes, it does look similar to the division sign so it’s really important not to get them confused.
2:49The root (or radical) sign is different from the division sign because,
2:53instead of having a curved front, its front shape is like a check mark.
2:58The number that you want to take the root of goes under the sign like this.
3:02So 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.
3:10In this case, you need to figure out what number you could multiply together a certain number of times to get 16.
3:16Ah - but there’s the catch! How many times?
3:19The answer we get from taking the root will depend on how many times that number would be multiplied together.
3:25But that would depend on the original exponent. So how do we know what that number is?
3:29Simple… the root symbol tells us.
3:31The root symbol actually includes the original exponent in it.
3:35What? You don’t see it?
3:37Oh… That’s because I didn’t draw it yet. And later in this video, you’ll understand why.
3:43So let’s put a little 2 right here above the check mark part of the root symbol.
3:48And 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.
3:57And, if you remember your multiplication table (or if you just look at our original example here)
4:02you’ll know that the answer to that is 4.
4:05Now do you see how the root operation is the inverse of the exponent operation?
4:09When doing the exponent, we asked, “What do we get if we multiply 4 together 2 times?”, and the answer was 16.
4:18But when we did the root operation, we asked, “What number could we multiply together 2 times to get 16?” and the answer was 4.
4:27Great! Now that you understand how exponents and roots are related, we’re going to look closer at how root operations work.
4:34To do that, we’re gonna change our root problem slightly. Let’s change the little 2 into a little 4.
4:41The first root was asking us to figure out what number we could multiply together 2 times to get 16.
4:48But this new root is asking us to figure out what number we could multiply together 4 times to get 16.
4:55That’s a bit trickier, huh? Can you think of a number like that?
4:58Yep, the answer is 2, because if you multiplied four ‘2’s together (2 × 2 × 2 × 2) you get 16.
5:07So the 2nd root of 16 is 4, but the 4th root of 16 is 2.
5:14Both those roots were pretty easy to figure out, right?
5:16But unfortunately, figuring out roots in math can be much harder.
5:20For example, what if we had this problem instead, “root 3 of 16”
5:25That means we need to figure out what number we could multiply together 3 times to get 16.
5:31Can you think of a number like that?
5:33[No] I can’t either!
5:35And unfortunately, it’s not easy to calculate what that number would be.
5:39Remember, even hough this look a little but like the division symbol, this is NOT just division!
5:45You can’t just divide 16 by 3 to get the answer. Roots are NOT the same as long division.
5:51So how DO we calculate a root like this?
5:54Well, there are special algorithms that you can use to calculate just about any root,
5:58but they’re kinda complicated, so we’ll save those for a future video.
6:03Instead, I’m gonna use a special root function on my calculator to get the answer.
6:08And on my calculator the button for that root function looks like this.
6:12To use it, I first enter the number that I want to take the root of, which is 16.
6:17Next, I hit the root function button, and then I enter 3 so it knows that I want the 3rd root of 16.
6:25Last, I hit the equals sign and voila… the answer is 2.519842… and the decimal digits just keep on going forever.
6:35Wow! See what I mean about roots being hard to figure out?
6:39That is a really complicated decimal number and you may even wonder if it’s the right answer. Well, let’s check…
6:45Based on what we know about exponents and roots, if we multiply this decimal number together 3 times, we should get 16, right?
6:53But to make it easier to check, let’s just round the number off to 2 decimal places. Let’s make it 2.52.
7:01If we multiply 2.52 together 3 times, (in other words if we take 2.52 to the 3rd power) we’ll get: 16.003.
7:12Well… that’s almost right. It’s really close to 16, isn’t it?
7:16The reason it’s not exactly 16 is that we rounded the number off which made it less accurate.
7:22But the more decimal digits we use, the closer we’ll get to 16.
7:26In math, the vast majority of roots are complicated number like this.
7:30And they’re hard to figure out unless you use a calculator or a special algorithm.
7:35That’s the bad news.
7:36But 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;
7:43...the ones that have nice whole number answers.
7:46And usually, you’ll only be asked to find 2nd or 3rd roots of numbers.
7:50Do you remember in the last video, we learned that 2 and 3 are the most common exponents.
7:55…so common in fact, that they even had special names.
7:59Raising a number to the 2nd power was called “squaring” it,
8:02and raising a number to the 3rd power was called “cubing” it.
8:06Well, it’s the same with roots. Since the roots 2 and 3 are the most common, they get special names also.
8:13The 2nd root is called the “square root”
8:16and the 3rd root is called the “cube root”.
8:19In fact, the “square root” is SO common that it’s basically the default root and its symbol even gets special treatment.
8:27Do you remember that when I first showed you the root symbol, I left out the index number that tells you what root to find?
8:33Well, whenever that number is left out, you can just assume that it’s 2.
8:38In other words, the root symbol, with no index number, is always the square root.
8:43So if you want someone to find a different root (like cubed or 4th or 5th)
8:48then you need to include that number so they know which root to find.
8:53And even though square roots are the most common, they’re not always easy to find.
8:57Most are still going to be big long decimal numbers, except for the “perfect squares”.
9:03It’s easier to find the square roots of the perfect squares because their answers can be found using the multiplication table.
9:10On the multiplication table, have you ever noticed that all the answers to problems
9:14where the same number is being multiplied together are on the diagonal of the table.
9:19In other words, 2x2=4,
9:223 × 3 = 9,
9:244 × 4 = 16,
9:255 × 5 = 25,
9:276 × 6 = 36, and so on.
9:31Well, those numbers are called the perfect squares because they’re the answers you get when you square a whole number.
9:37And that means if you take the square root of a number along that diagonal, you get a nice whole number as your answer.
9:44The square root of 4 is 2.
9:47The square root of 9 is 3.
9:49The square root of 16 is 4.
9:52The square root of 25 is 5, and so on.
9:55See what I mean?
9:57Those roots are really common and they’re also easy to figure out if you know your multiplication facts.
10:03So if you’re new to exponents and roots, learning the perfect squares is the place to start.
10:08Once you understand how those exponents and roots work, you’ll be ready to figure our tougher problems.
10:13Alright… so now you know how exponents are related to roots. They’re inverse operations and they undo one another.
10:21And you also know that, just like 2 and 3 are the most common exponents,
10:25the square root and the cube root are the most common roots.
10:30You also know that finding roots is usually not very easy.
10:33That’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.
10:39You’re not alone! We think it’s hard too and would normally just use a calculator to find them.
10:45The good news is that some roots are easy to find, like the perfect squares, so be sure to focus on learning them first.
10:52And remember, to get good at math you need to actually practice what you learn from watching videos,
10:57so be sure to do some exercise problem.
10:59And as always, thanks for watching Math Antics and I’ll see ya next time.
11:04Learn more at www.mathantics.com