Do you know why I pulled you over today?
Uh… some law?
That’s right a law!
This law to be exact!
Laws of Exponents!?
I’ve never even heard of these!
How was I supposed to know?
Well, you must have never heard of Math Antics.
They’re this really cool math video series. They have all sorts of basic math videos…
…and the host is REALLY funny!
Is this some sort of ad? Can I skip this?
No, I think you’ll really like it. Check it out…
[Intro music]
Hi, this is Rob. Welcome to Math Antics.
In this video we’re gonna learn about the Laws of Exponents.
If you look up the laws of exponents online or in a math book
you’ll probably see a long list of equations that look something like this.
Wow! That’s kinda overwhelming when you see them all at once!
But don’t worry… we’ll take them one step at a time and you’ll see they’re not that complicated after all.
But before we get going, if you’re not confident with the basics of exponents,
I highly recommend watching our previous videos about them before moving on in this video.
Okay, let’s start with just the first two laws on our list,
which should look familiar if you watched our video called “Exponents in Algebra”.
They’re just the two “rules” we learned in that video,
ya know… rules…laws… same difference.
So you probably already know them!
They simply tell us that anything raised to the first power is itself,
and anything raised to the zeroth power is just 1.
And because we already know about exponents that are higher integer values
like ‘x’ to the 2nd and ‘x’ to the 3rd,
that means we’ve pretty much got things covered, right?
Ah… not so fast!
Don’t forget that integers can have negative values too.
For example, what if we had the expression ‘x’ to the power of negative 1 ?
...or negative 2 ?
...or negative 3 ?
We know that exponent are a way of doing repeated multiplication,
but how in the world could you multiply something together a NEGATIVE number of times?
Well… you can’t.
Fortunately, the next law on our list tells us how to interpret a negative exponent.
That law says: ‘x’ to the negative nth power equals 1 divided by ‘x’ to the nth power.
And if you think about it, that kinda makes sense.
A negative number is the ‘inverse’ of its positive counterpart,
and division is the inverse operation of multiplication, right?
So a negative exponent is basically repeated division.
‘x’ to the negative 1 would be 1 divided by ‘x’
‘x’ to the negative 2 would be 1 divided by ‘x’ divided by ‘x’
‘x’ to the negative 3 would be 1 divided by ‘x’ divided by ‘x’ divided by ‘x’, and so on…
Seeing it like this makes the pattern clear,
but mathematicians prefer to express negative exponents in fraction form where
1 is divided by the same number of ‘x’s multiplied together.
But since those multiplied ‘x’s are all on the BOTTOM of the fraction,
you’re actually dividing by all of them.
Here’s an example that will help you see that that’s true: 2 to the negative 3rd power.
Let’s first try that as a repeated division problem like our ‘pattern’ shows us.
We always start with a 1 so we’d have: 1 divided by 2 divided by 2 divided by 2.
If we do those operations from left to right using a calculator, we get 0.125 as the answer.
Now let’s write it fraction form like our law of exponents tells us we can.
2 to the negative 3rd power would be the same as 1 divided by 2 to the 3rd power.
And that’s the same as 1 over 2 times 2 times 2.
Which simplifies to 1 over 8. And 1 over 8 simplifies to 0.125.
See! Whether you write it as a pattern of repeated division
or in fraction form like our ‘inverse law’ shows, you get the same answer.
And now you know how to handle any expression with a negative exponent…
it’s just 1 over the same expression with a positive exponent.
‘x’ to the negative 1 is 1 over ‘x’ to the positive 1 (or just 1 over ‘x’)
‘x’ to the negative 2 is 1 over ‘x’ to the positive 2
‘x’ to the negative 3 is 1 over ‘x’ to the positive 3, and so on…
Alright… three laws down, five more to go.
And these next five show us how we can do various math operations involving exponents.
In fact, the next law tells us how we can take a number raised to a power and then raise THAT to a power.
As you can see, it shows an expression (‘x’ to the power of ‘m’) grouped inside parentheses,
and then that whole group is being raised to the nth power.
It’s a nesting situation, kinda like those Russian nesting dolls.
So what if someone asked you to simplify the expression ‘x-squared’ cubed?…
which means the entire’x-squared’ term is raised to the 3rd power.
Well our law tells us that we can simplify that by multiplying the exponents together.
…see how it equals ‘x’ to the power of ‘mn’ (which means ‘m’ TIMES ’n’)
That means ‘x-squared’ raised to the 3rd power would be the same as
‘x’ to the power of 2 times 3, which is 6.
…want to see why that’s true?
Well think about what it would mean to raise ‘x-squared’ to the 3rd power.
It would mean multiplying THREE ‘x-squared’ terms together like this.
And each one of those ‘x-squared’ terms simplifies to ‘x’ times ‘x’, right?
So we end up with 6 ‘x’s being multiplied together, which is just ‘x’ to the 6th power.
See… our law works great!
If you have a number raised to a power, and that’s all raised to another power,
you can just multiply the two exponents together to simplify it.
And it works for negative exponents too.
Like what if we had ‘x-squared’ raised to the negative 3rd power.
Our law tells us that that’s the same as ‘x’ to the power of 2 times negative 3
which is ‘x’ to the negative 6th.
To see if that’s true, we’ll need to use the law we just learned about negative exponents
and re-write this as 1 OVER ‘x-squared’ to the positive 3rd power.
That simplifies to 1 over ‘x-squared’ times ‘x-squared’ times ‘x-squared’,
which in turn simplifies to 1 over 6 ‘x’s being multiplied together.
And that all checks out because 1 over ‘x’ to the 6th would be the same as
‘x’ to the negative 6th power.
Pretty cool, huh?
Okay… we’re half way through our list of laws… and we’re gonna look at the next two as a set
because they tell us how we can multiply and divide expressions that have the same base.
And that’s important because we couldn’t simplify them to have a single exponent if the bases were different.
The first law says that if we have the base ‘x’ with exponent ‘m’
being MULTIPLIED by the SAME base ‘x’ with exponent ’n’,
we can combine them simply by ADDING the exponents together.
And the second law says if we have the base ‘x’ with exponent ‘m’
being DIVIDED by the SAME base ‘x’ with exponent ’n’,
we can combine them simply by SUBTRACTING the exponents.
Let’s see some examples of each… like this one:
2 to the 3rd times 2 to the 4th.
Does that fit the pattern of the first law?
Yep, the base of both expressions is the same, but they happen to have different exponents.
The law would still work if the exponents were the same,
but they don’t have to be… just the bases have to be the same.
Our law tells us that this would equal 2 to the power of 3 plus 4, or 2 to the 7th power.
But does it? Well, let’s break it down and see…
2 to the 3rd is 2 * 2 * 2 and that’s being multiplied by 2 to the 4th which is 2 * 2 * 2 * 2.
That’s a lot of twos being multiplied together… 7 twos to be exact.
Ah ha… so that IS what you’d get by just adding the exponents together since 3 + 4 = 7.
And this law makes total sense if you think about what an exponent really means.
The exponent is telling you to do repeated multiplication of the base, right?
So this first part is telling you multiply 3 twos together
and this second part is telling you multiply 4 twos together.
So that’s why you can add the exponents together IF the base is the same.
Or think about it like this… if we had 10 ‘x’s all being multiplied together,
we could form different groups of them and combine them using exponents.
Like we could combine the first 4 ‘x’s into ‘x’ to the 4th
and combine the remaining 6 ‘x’s into ‘x’ to the 6th.
And of course, those expressions would be multiplied together since all of the 'x’s were being multiplied.
But you’d probably never wanna do that, would you?
I mean… why not just combine all 10 ‘x’s into the expression ‘x’ to the 10th?
Ah… but there you see that our law holds true.
‘x’ to the 4th times ‘x’ to the 6th WOULD equal ‘x’ to the power of 4 plus 6 (or ‘x’ to the 10th).
Okay, now let’s move on and see some examples of the second law in this set
which tells us how to divide expressions with the same base.
Suppose we have the expression 5 to the 3rd power divided by 5 to the 2nd power.
Our law says that we can simplify this but subtracting the exponents.
Specifically, we take the exponent on the top and subtract the exponent on the bottom from it.
If we do that, the simplified version would be 5 to the power of 3 minus 2, or 5 to the 1st power.
But is that right?
To see, let’s write the expression out in expanded form.
On the top of our division problem, we have 5 to the 3rd which is 5 * 5 * 5
and on the bottom we have 5 to the 2nd which is 5 * 5.
Does this look like something you’ve seen while simplifying fractions?
Yep. Since all the bases are the same,
they form pairs of common factors on the top and bottom that can be canceled out.
This 5 over 5 cancels and this 5 over 5 cancels.
If you don’t know why that works, be sure to watch our video about Simplifying Fractions.
And what do we end up with?
Well, all the factors on the bottom canceled out, which leaves 1 since there’s always a factor of 1.
And there’s only one 5 left on the top.
So our expression simplified to 5 over 1 or just 5.
Following our law, we got a simplified version of 5 to the 1st power, which is also just 5.
So it really did work! But to make sure you’ve really got it,
let’s try using this law again with the expression ‘x’ to the 4th power over ‘x’ to the 6th power.
Hmmm… this one is interesting.
Our law says that we can simplify it by subtracting the bottom exponent from the top, right?
But in this case, that will give us a negative exponent because the bottom exponent is bigger than the top.
4 minus 6 would be negative 2, so according to our law,
this expression should be equal to ‘x’ to the negative 2.
Let’s try writing it out in expanded form to see if that’s true.
On the top, ‘x’ to the 4th would be the same as 4 ‘x’s multiplied together,
and on the bottom, ‘x’ to the 6th would be the same as 6 ‘x’s multiplied together.
Once again, we see that there are pairs of common factors that we can cancel…
four pairs to be precise.
And when we cancel them all, we’re left with a 1 on the top (cuz there’s always a factor of 1)
and only 2 ‘x’s multiplied together on the bottom.
If we re-combine these two ‘x’s we get 1 over ‘x squared’.
And if you remember the law we learned earlier about negative exponents,
you’ll see that 1 over ‘x-squared’ is exactly the same as ‘x’ to the power of negative two.
So these laws really do work!
Okay, it’s finally time to look at the last two laws on our list,
and fortunately, they’re pretty easy ones, so we’re not gonna spend too much time on them.
These laws look kinda similar to the last pair.
Just like before, the first one involves multiplication and the second one involves division.
But notice that in these laws, the bases are different, but the exponents are the same.
That’s the exact opposite of the situation with the last pair of laws.
It turns out that these laws aren’t about simplifying exponents,
they’re about how you can distribute (or un-distribute) a common exponent to different bases.
The fist law shows this group (‘x’ times ‘y’) that’s being raised to the power of ‘m’
…and it says that you can re-write it as ‘x’ to the ‘m’ TIMES ‘y’ to the ‘m’.
In other words, you can ‘distribute’ the exponent to each factor in the group.
And the second law shows the group (‘x’ divided by ‘y’) that’s being raised to the power of ’n’
…and it says that you can re-write it as ‘x’ to the ’n’ DIVIDED BY ‘y’ to the ’n’.
In other words, you can ‘distribute’ the exponent to each part of the fraction.
Of course, these laws would work in revers too and you could ‘un-distribute’ the exponents if they’re the same.
For example, if you're given the expression ‘x-squared’ times ‘y-squared’,
you could re-write that as the quantity ('x' times 'y') squared.
And if you had the expression ‘x-squared’ divided by ‘y-squared’,
you could re-write that as the quantity (‘x’ over ‘y’) squared.
Here are two expressions that will help you see why you can distribute
(or un-distribute) exponents like our laws show.
In the first expression, we have the quantity ‘x’ times ’y’ squared.
That’s the same as ‘x’ times ’y’ times ‘x’ times ’y’.
And the commutative property says that we can rearrange those factors like this:
‘x’ times ’x’ times ‘y’ times ’y’.
But look... we can simplify that into ‘x-squared’ times ‘y-squared’.
So that checks out.
In the second expression, we have the quantity ‘x’ over ‘y’ squared.
That’s the same as ‘x’ over ‘y’ times ‘x’ over ‘y’.
To multiply these fractions we just multiply the tops and multiply the bottoms
which give us ‘x’ times ’x’ over ‘y’ times ’y’.
And that simplifies to ‘x-squared’ over ‘y-squared’.
So that one checks out too.
Alright, so now you know about the so-called Laws of Exponents.
And there’s a good chance that you’ll see them explained in slightly different forms,
or different orders, or even using different terminology in other math videos or books.
but the basic ideas will be the same.
Some people like to try to memorize this list of laws, and you can do that if you want to…
but it’s an even better idea to focus on knowing how exponents really work,
because if you truly understand that, you can actually figure out a lot of these laws for yourself.
And what’s the best way to understand how exponents really work?
Yep, ya gotta practice! So be sure to do some problems with exponents on your own.
As always, thanks for watching Math Antics and I’ll see ya next time.
Learn more at www.mathantics.com
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