>> Math is cool and you can do it.
>> All right. We're covering the power rule for exponents.
So here's an example, we have something in parentheses raised to the power of 4.
So, this means x cubed times x cubed times x cubed times x cubed.
And then of course you could break that down into 3 factors of x, 3 more factors,
3 more and 3 more or you could use the rule for the product and add the exponents.
In any case, you should get x to the 12th.
So the question is, is there a quicker way to get that answer?
And yes, there is.
You can multiply the exponents.
So that's all there is to the power rule for exponents.
Again, if you forget it, you could always break it down the long way as shown first.
Now, it's better to do that than write x to the 7th and just guess for instance.
You can also note that there's a parentheses right next to the 3 and 4 and usually,
when you see a number next to another parentheses, you multiply it.
So that could be a little hint, maybe you're going to multiply right there.
Right? I'm just giving you ideas.
So for instance, look at the difference between x cubed times x to the 4th, no parentheses,
and kind of room in your mind to think, "Hey, if I wanted,
I could think if there's a little plus sign between there."
There really is not a plus sign, I'm just saying in your mind, you could imagine there's room
for a plus sign so that's x to the 7th.
But if you've got x cubed, raised to the 4th power,
you have a parentheses here that might help remember that.
Help you remember that's when you multiply.
Remember, there really is not a plus sign there.
Okay, so how about this one?
What if you had a to the 12th to the 10th power?
You could do that the long way as well.
You could write out a to the 12th 10 times, multiply it out and add
or the simpler way would be simply to multiply 12 times 10 and get a to the 20th.
This rule still applies even if the base is a number.
So consider 6 cubed to the 8th power, the base is still 6 and you multiply 3 times 8 to get 24.
Okay, let's look at this example.
We're going to develop the power of a product or quotient rule.
So what this really means is 2x cubed over m squared times 2x cubed over m squared,
times another 2x cubed over m squared times another 2x cubed over m squared.
So you certainly don't need a new rule, you simply write out what it means
to raise something to the 4th power and then multiply the numerators
and multiply the denominators using the laws of exponents.
So in the numerator I have 2 times 2 times 2 times 2 or 16.
I am going to have x to the 12th power, because you have to add the exponents there,
sorry for that technical difficulty.
So we're going to have x to the 12th and m to the 8th in the denominator.
The question is, is there a faster way to do this?
And yes. What you can do is everything inside could be raised to the 4th power.
So, remember, this is 2 times x cubed so it's a really a 2 to the 4th power.
You don't need to put a parentheses around the 2, but you can.
You will need to put the parentheses around x cubed.
And you will need to put a parentheses around m squared.
And you're raising each product or quotient to the 4th power, and now you can use your law
of exponents for the power rules, 2 to the 4th is 16,
x cubed to the 4th power you would multiply exponents and m squared
to the 4th power is m to the 8th.
So, notice I get the same answer.
Here's one for you to try.
So put it on pause, try it and then come back.
Okay, so let's go through this.
Now, there's two ways of thinking about this negative 5.
You could think of it as a negative 1 times 5 or the number negative 5.
In any case, you're going to have a negative 5 squared, if you think of it as a negative 5.
And then you've got a x cubed squared and then you have a n squared.
Now, I could just write n squared.
I don't need to put it in parentheses.
That's up to you, it doesn't hurt.
And we have n squared to the 8th.
So negative 5 to the 2nd power, remember, means negative 5 times negative 5.
That's going to be 25.
Multiplying exponents for the next one, 3 times 2 is 6, n squared,
multiplying exponents 8 times 2 for the denominator, is n to the 16th.
So how did you do?
I hope you got it correct.
Okay, here's a much more complicated one.
Let's see if you can do this problem.
So we're going to have to use more than one rule.
So, put it on pause, see if you can get it all the way through.
Okay. Now, I need to first do my exponents here so I need to take care
of this and also at the bottom, this.
Okay, so what does that give me?
We still have the negative 4, x to the 5th m
and then I've got 3 squared, could you do that in your head?
That's fine or you could write 3 squared.
Then I've got x squared squared and I've got m squared.
That's going to be in the numerator.
And in the bottom, I'm going to have a 2 cubed and an x cubed.
Okay so what do I have at the top here?
Got negative 4 x to the 5th, if you want you could think of this as all times.
M times 9 times x to the 4th times m squared and at the bottom, I've got 8 times x cubed.
Now, it's true that you might be able to do some canceling right now,
but I'm going to simplify the numerator and denominator first.
Actually, I'm going to cancel that 4 and the 8.
That's going to make things a little bit easier.
Since there's only one negative sign my answer's going to be negative.
I'm going to write that right out in front.
So I could cancel then the 4 and the 8, so I get a 2.
Okay. Now, in the numerator, the only number, whole number I have here left is a 9.
Okay so take care of the 9.
Now, I've got my x to the 5th and I've got my x to the 4th.
That will give me x to the 9th.
I also have m and m squared.
Remember that means m to the 1st power, so that's going to give me m cubed.
And of course, I also have the 9, right?
So I've pretty much taken care of everything in that numerator.
Now, what I have left on the denominator is a 2 and a x cubed.
[ Pause ]
Okay. But are we done?
We can't cancel the 9 and 2.
There's no m's in the numerator and denominator, but I do have a x to the 9th over x cubed.
So here's where I'm going to use the quotient rule and subtract my exponents,
9 minus 3 so that my final answer here, have the minus sign and if I subtract in the numerator,
I'm going to get x to the 6th m cubed all over 2.
So wow. If you did that one correctly, you have got it.
[ Pause ]
Let's do one more.
Hmm, negative 7 to the 22nd over negative 7 squared.
This is just like x to the 22nd over x squared, except instead of x and I have a negative 7.
So you subtract exponents, 22 minus 2 is, oops.
I meant to write 20.
So I have negative 7 to the 2nd power
and that will give me negative 7 times negative 7, simplifies to 49.
Okay. So, we've done quite a few different types of problems here.
We're using the product rule, and then the power of a product rule
and then we're putting it all together.
We used the product rule, the quotient rule, the power rule,
the product of the quotient or a power rule.
Lots to think about.
Just make sure that if you're not sure about the rules, then you just write it out the long way.
If you know what x squared means x times x, simply write it out like that.
Okay, have a good time.
Let's summarize the power of a product rule.
If you have something in parentheses raised to some exponent then you have to take each factor,
in this case you have a factor of a and a factor of b in the parentheses.
and each one gets raised to the exponent.
For a power of a quotient rule, if you have a quotient raised to some power,
then you have whatever's in the numerator raised to that power and whatsever
in the denominator raised to that power.
A quick example might be 2 n to the 4th raised to the 3rd power.
There's really 2 factors.
There's a 2 and n to the 4th.
So we have 2 cubed and n to the 4th cubed, which is 8 n to the 12th after using the product rule,
here where you multiply 4 times 3.
Okay, have fun.
Be careful.