Exponents, also known as indices, are a mathematical operation representing repeated multiplication of a base number by itself a specified number of times, indicated by a smaller superscript number.
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Hi! Welcome to Math Antics.
In this lesson, we’re going to learn about an important math operation called exponents.
…sounds kinda weird, huh?
Indeed it does my good man.
And I’m afraid it’s because you’re pronouncing “exponents” incorrectly.
The proper pronunciation is “Indices”. “In…di…ces”.
Oh yeah, that’s right. In a lot of countries, exponents are called Indices.
But the good news is that the concept is the same no matter what term is used.
And, since I’m usually an American, I’ll usually use the term “exponents” in this video,
but I’ll occasionally point out the other terminology too.
Sound good to you?
Jolly good sir! Toodle pip!
Alright then… but what are exponents (or indices)?
Well, just like the four main arithmetic operations that we use in math,
exponents are a type of math operation.
They tell us what to DO with numbers.
Ok, but what do they tell us to do?
Exponents tell us to take a number and multiply it by itself a certain number of times.
In other words, exponents are basically repeated multiplication.
To help you understand what I mean, let’s review multiplication for just a second.
Do you remember how multiplication is basically repeated addition?
For example, 2 × 4 is a shorthand way of writing 2 + 2 + 2 + 2.
It’s the same as four ‘2’s being added together which equals a total of 8.
Now, let’s compare that to a similar exponent.
Yep, that’s an exponent, and in a minute we’ll talk about why it looks like that
and why the numbers are different sizes.
But for now, you just need to know that this exponent tells us to take the number 2 and multiply it 4 times.
In other words, you’d take four ‘2’s and multiply them together.
2 × 2 × 2 × 2 which would equal 16.
So, multiplication shows repeated addition and exponents show repeated multiplication.
And we’ll see more examples of how exponents work in a minute.
But first, let’s talk a little bit about why exponents look the way they do.
Exponents involve two numbers.
The first number is the one that gets multiplied repeatedly a certain number of times.
And it’s written full size and it’s called the “base”.
The second number tells us how many copies of the base to multiply together.
It’s written much smaller and up at the top of the line like this.
And it’s called the exponent (or the Index)
And when reading exponents, some interesting terminology is used…
This exponent would usually be read, “two to the fourth power”, or just “two to the fourth” for short.
And this exponent would be read, “three to the fifth power”, or just “three to the fifth”.
And this exponent would be read, ”ten to the sixth power”, or just “ten to the sixth”.
…get the idea?
Oh, and you’ll often hear people say that a number is “raised” to a certain power,
…ya know… cuz it sounds kinda cool.
But it’s just another way of saying the same thing.
Unfortunately, that terminology has created a little confusion…
Traditionally, you’d say you have a base raised to a number called an exponent,
and the answer you get by doing that is called a “power” of that base.
But when people started saying things like “2 to the 4th power” or “2 to the power of 4”,
it made it sound like the exponent was also called the power.
The result is that nowadays, you may hear people use the words “exponent” and “power” to mean the same thing.
It’s unfortunate that the terms have become so “mixed up” or “wonky”,
but as long as you understand that the base is the number that gets multiplied repeatedly
and the exponent tells you how many of them to multiply, you’ll be in good shape.
But, I know what some of you are thinking.
“Where’s the operator symbol?”
If exponents are math operations, don’t we need a symbol that goes between them,
like there is with multiplication and the other arithmetic operations?
Well, yes and no.
Most of the time, we don’t need a special operation symbol because of the way the numbers are written.
Since the exponent is written smaller and up at the top of the line,
it looks much different from a normal digit, so we don’t need to use a symbol.
In math, whenever you see a regular sized number with a smaller number up and to the right, you know it’s an exponent.
And once you get used to seeing exponents like this, it’s easy to recognize them.
The only real concern is when writing exponents down on paper when you’re trying to solve problems.
If you’re not careful, or have really messy handwriting,
you might accidentally confuse an exponent (like two to the fifth)
with a two-digit number (like 25) and obviously that would be a problem.
But even if you’re careful when writing exponents, there are some times when you really do need a special symbol,
like when you type an exponent into a computer.
In that case, it’s very common to use the “carrot” symbol as the exponent operator.
The carrot symbols looks like this.
So, 2 to the 5th power would be written as “2 carrot 5”,
and 3 to the 4th power would be written as “3 carrot 4”.
And that notation is used all the time in computer programing.
Oh, and one more thing you should know before we move on…
is that exponent operations do NOT have the commutative property.
In other words, you can’t switch the order of the numbers, without getting a different answer.
For example, 2 to the 5th power is 2 × 2 × 2 × 2 × 2 which equals 32.
But if we switch the numbers: 5 to the 2nd power is 5 × 5 which equals 25.
So exponents do NOT have the commutative property.
Okay, now that you know what exponents are (or indices if you prefer) and you know how they’re written,
it’s time to talk about how they’re used in math.
As I mentioned earlier, exponents are a way of doing repeated multiplication.
3 to the 2nd power is the same as 3 × 3
3 to the 3rd power is the same as 3 × 3 × 3
3 to the 4th power is the same as 3 × 3 × 3 × 3
3 to the 5th power is the same as 3 × 3 × 3 × 3 × 3
…see the pattern?
So exponents can save you a lot of writing when you need to show repeated multiplication.
But, what about when you actually need to DO repeated multiplication.
Do exponents help then?
Well… yes and no.
If you have to actually figure out what 3 to the 5th power is, you still need to multiply '3' together 5 times.
And you could do that by hand, or you could use a calculator to help you.
In fact, if you have a multi-function or ‘scientific’ calculator,
it might have a button on it that looks like this: ‘x’ to the power of ‘y’,
which makes calculating exponents really easy.
Let’s try using that calculator function to figure out what this exponent would be:
6 to the 4th power (which is the same as four ‘6’s multiplied together).
On most calculators, this is how the exponent button works…
First, you type in the number that’s the base of your exponent. So in this problem, we’d type in 6.
And then you’d hit the exponent button,
which might not do anything besides let the calculator know that the next number you give it will be an exponent.
So next, you enter the exponent which is 4 in this case.
Now all you have to do is hit the equals sign to get the answer which is 1,296.
Wow! As you can see, knowing how to do that with a calculator saved us a lot of work.
And now we know that four ‘6’s multiplied together will be 1,296.
In the examples we’ve seen so far, we’ve only had simple one-digit exponents like 2, 3, or 4.
But exponents can be any number. You could have 2 to the 84th power, or 12 to the 516th power.
In fact, as crazy as it sounds, you can even have exponents that are decimals or negative numbers!
But we’ll save advanced exponents like that for future videos.
In the last part of this video,
I want to focus on the two most common exponents you’ll encounter, which are 2 and 3.
That is, you’ll very often see a number raised to the second power or to the third power.
In fact, those exponents are SO common that they even have special names.
When a number is raised to the second power, we say it’s being “squared”.
So 5 to the second power is also referred to as “5 squared” and 12 to the 2nd power would be “12 squared”.
Now if you’re new to exponents, that might sound kinda funny,
but can you think of a reason why the term “squared” is used?
Yep! That’s how you’d calculate the area of a square shape.
Since squares have sides that are all the same length, if you multiply that length together twice
(like 4 × 4 or 5 × 5 or 6 × 6) it gives you the area of that square.
And it’s a similar story for numbers that are raised to the third power.
In that case, you’d say that the number is being “cubed”.
So, 5 to the third power is also “5 cubed”, and 8 to the 3rd power would be “8 cubed”.
Since cubes are 3-dimensional objects with sides that are all the same length,
if you multiply that length together 3 times, you’d get the volume of the cube.
So squaring a number gets its name from squares and cubing a number gets its name from cubes.
And that helps explain why the exponents 2 and 3 are really common in math.
Because we interact with 2 and 3 dimensional objects all the time in the real world,
there are lots of practical applications.
Okay, so now you know that exponents are a way of showing repeated multiplication,
and you also know what the two numbers in an exponent mean.
The base is the number that will get repeatedly multiplied together,
and the exponent (or index) tells us how many times to repeat it.
In the next video, we’ll learn more about exponents and their inverse operations called roots.
Learning math takes a lot of practice, so be sure to practice what you’ve learned in this video.
Thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com
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