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Lecture Comments (3)

2 answers

Last reply by: sherman boey
Sat Aug 23, 2014 4:32 AM

Post by Amanda Black on December 8, 2013

On example 6 for the second problem, shouldn't it be just y to the fourth instead of y to the fourth and then raised to the fourth again? I got -27 x to the 11th and y to the 10th.

Integer Exponents

  • The product rule for exponents tells us that we may add the exponents of two expressions that are being multiplied together, as long as the bases are the same.
  • The power rule tells us that we may multiply the exponents together if we have an expression with an exponent, raised to another power.
  • The power rule can also be applied to multiplication and division. In these rules the exponent is given to each of the parts that are multiplied, or divided. Careful, the power rule does not allow us to apply exponents over addition or subtraction.
  • The quotient rule for exponents tells us that we may subtract exponents of two expressions that are being divided, as long as the bases are the same.
  • The negative exponent rule says that if an expression is raised to a negative power, it can be written as 1 divided by the same expression with a positive power. As a short cut, think of taking expressions with negative powers and switching their location.
  • An expression raised to the power of zero is equal to one.

Integer Exponents

  • 12a3 + 2b5 + 2c4
  • 36x7 + 6y3 + 1z
  • 30m1 + 2n3 + 7
  • (4)3(a4)3(b9)3(c2)3
  • 64a4 ×3b9 ×3c2 ×3
  • (8)2(x2)2(y3)2(z4)2
  • 64x2 ×2y3 ×2z4 ×2
  • (5)4(g3)4(h6)4(i)4
  • 625g3 ×4h6 ×4i4
  • (7)2(a2)2(b3)2(c3)2
  • 49a2 ×2b3 ×2c3 ×2
(3g5h3i4)2( − 4g2h6i7)3
  • [(3)2(g5)2(h3)2(i4)2][( − 4)3(g2)3(h6)3(i7)3]
  • 9g10h6i8 ( − 64)g6h18i21
  • (9)( − 64)(g10 ×g6)(h6 ×h18)(i8 ×i21)
  • − 576(g10 + 6)(h6 + 18)(i8 + 21)
- 576g16h24i29
(5g5h6i7)3( − 8g2h3i4)2
  • [(5)3(g5)3(h6)3(i7)3][( − 8)2(g2)2(h3)2(i4)2]
  • 125g15h18i21(64)g4h6i8
  • (125)(64)(g15 ×g4)(h18 ×h6)(i21 ×i8)
  • (125)(64)(g15 + 4)(h18 + 6)(i21 + 8)
( − [1/3]x2y5z6 )2( [5/6]x4y7z3 )3( x5y4 )4
  • ( − [1/3] )2(x2)2(y5)2(z6)2( [5/6] )3(x4)3(y7)3(z3)3(x5)4(y4)4
  • [1/9]( x4y10z12 )[125/216]( x12y21z9 )( x20y16 )
  • [1/9] ×[125/216]( x4 ×x12 ×x20 )( y10 ×y21 ×y16 )( z12 ×z9 )

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.


Integer Exponents

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Objectives 0:09
  • Integer Exponents 0:42
    • Exponents 'Package' Multiplication
  • Example 1 2:00
  • Example 2 3:13
  • Integer Exponents Cont. 4:50
    • Product Rule for Exponents
  • Example 3 7:16
  • Example 4 10:15
  • Integer Exponents Cont. 13:13
    • Power Rule for Exponents
    • Power Rule with Multiplication and Division
  • Example 5 16:18
  • Integer Exponents Cont. 20:04
  • Example 6 20:41
  • Integer Exponents Cont. 25:52
    • Zero Exponent Rule
    • Quotient Rule
    • Negative Exponents
    • Negative Exponent Rule
  • Example 7 34:05
  • Example 8 36:15
  • Example 9 39:33
  • Example 10 43:16

Transcription: Integer Exponents

Welcome back to

In this lesson we are going to take a look at integer exponents.0002

What does it mean? You take a quantity and raise it to a power.0005

A lot of other things we will also be picking up along the way.0010

Let us get a good overview.0013

We will start off by interpreting what it means to raise something to a power when that power is a nice whole number.0015

We will see that we can often combine things using the product and the power rule because all these things have the same base.0021

We will see what it means to raise things to a 0 power and get into the quotient rule.0028

And get into understanding what happens when you raise something to a negative power.0034

Watch for all of these things to play a part.0038

One way that we can look at multiplication is that it is a packaging up of addition.0046

I have the value of x and we know that I have added together 5 times.0053

Rather than trying to string out to x + x + x over and over again I can write that same thing by just saying 5 multiplied by x.0059

It is a great way to take a long addition problem and package it all down into a nice single term. 0074

Exponents are the things that we are interested in as long as we are dealing with whole numbers you can look at that as packaging up some multiplication. 0081

Here I have those x’s once again, I have 5 of them all being multiplied together but rather writing all that, I'm going to use my exponents.0090

Specifically the bottom here will be our base. 0105

Remember, that is what we are multiplying over and over again.0108

The exponent itself tells us how many times.0111

Let us get some practice with just understanding how exponents work before moving on. 0121

With this, I simply want to write them into their exponential potential form.0128

If I have 5 × 5 × 5, I can see that it is the 5 being repeatedly multiplied and I did it 3 times.0132

I could consider this one like 53.0139

If I wanted to take this one little bit farther, this is equal to 125.0143

You have to focus on what is being multiplied over and over and over again to properly identify your base.0150

In this one, we have -2 × -2 × -2, we can see that it is being multiplied by itself 4 times.0157

We want the entire value of -2 multiplied by it self 4 times so we will use an exponent of 4.0167

If we had to simplify this one, I have -2 × -2 × -2 × 2 and that would be 16.0177

Keep an eye on the base, so you know what is being multiplied and the exponent will tell you how many times to do that.0186

In this next two, we just want to evaluate the exponential expression.0196

Let us see if we can identify the base here.0200

This is a 34 that happens to be negative sign out front and my next example, it is almost the same thing but it is -34.0203

Watch how I treat both of these a little bit differently. 0212

In this first one, the only thing that is considered the base is actually that number 3, 3 × 3 × 3 × 3 multiplied by itself 4 different times.0215

What should I do with that negative sign out front, it is still out front, but since it is not part of the base and it only shows up once.0228

I can take care of it from here, 3 × 3 × 3 × 3 a lot of 3’s in there, but that will equal 81.0235

Of course that negative sign is out front along for the right.0244

In the next one, these parentheses are highlighting that the entire -3 is what is in the base so I'll repeat the entire -3. 0248

Look at this one, I'm dealing with a -3 multiplied by itself 4 times.0267

I know that a negative × negative would be positive and another negative × negative would be a positive, 0272

the answer to this one action turns out to be 81.0278

Use those parentheses to help you know what is in the base and it is good for all of those negatives.0283

One of our first rules that we can throw into the mix is the product rule for exponents.0293

This will help us when we have two things and both of them are raised to an exponent.0299

It is a great way to actually put them together so they only have one thing raised to an exponent.0306

The way this rule works is, we want to make sure that both the bases are exactly the same.0311

I have a and a as my base, if they are the same then it says we simply need to take each of their exponents and add them together. 0318

It seems like a little bit of an odd rule how you can always start with multiplication and end up with addition 0327

but let me show you an example of why this works.0333

I’m going to base that off just what you know about exponents that it is repeated multiplication. 0336

Here I have 24 power be multiplied by 23, let us pretend I knew nothing about the product rule.0342

I can start interpreting this by using repeated multiplication, so 24 will be 2 × 2 × 2 × 2 until I done that 4 times in a row.0350

I can do the same for 23 that would be 2 × 2 × 2.0364

You can see what we have here, the whole bunch of 2s are all being multiplied together and there is even no real need to put on those parentheses.0373

I will just write 2 × 2 × 2 until I got all of them written down. 0381

As long as I have all of these 2s written out, it was said earlier that exponents are just repeated multiplication, 0388

I can actually package these all backup.0394

If you count how many are here, it looks like we have 7 total.0398

I can say that this is 27.0404

Now comes the important part, if you look at our original exponents that we have used the 4 and 3 and you add them together 0409

you can see that sure enough, it does add to 7.0418

The reason why this is happening is because you are simply throwing in a few more numbers to multiply by and incrementing up that exponent.0421

That is why the product rule works.0431

Let us go ahead and practice with it now.0434

In our first one I have -75 × -73.0439

The first thing you want to recognize is both of those bases are the same, so we will just focus on their exponents the 5 and 3.0445

This one as -7^(5+3) and then I can go through and add those 2 things together, -78.0456

You could simplify it further from there if you want to but I’m just going to leave it like that so you can see the product rule in action.0472

The next one will involve a little bit more work and that is why I put in the -4 and 3.0479

Before I can get too far, I’m going to start rearranging the location of the 4 and the p's raised to the exponents.0488

What allows me to do this is the commutative property for multiplication that the order of multiplication does not matter.0495

This is just helping me organize things a little better so that I have my numbers all in one spot and my exponents on one spot as well.0504

That looks good.0513

Put the -4 and 3, I will just straight multiply those two together and get -12.0515

With the other two, I can see that they are both the same base of p.0521

I will look at adding their exponents together so p^(5+8).0526

Here is -12 p13 not bad.0535

One last, let us see if we can use the product rule in that one.0542

Looking at this one you will know that they both have a base of 6.0546

One is raised to the 4th power and one is raised to the power of 2. 0549

But there is a slight problem with this one.0553

This one is dealing with addition and the product rule, the one that we are interested in is for multiplication.0557

Since the product rule only applies to multiplication, it means we can not use the product rule here.0570

Even though that my bases are the same it simply not going to work here. 0584

If I did want to continue this one out, I would not use the product rule. 0589

I would just simply take 64 and see what value that is.0593

I will get 1296 then I will get 62 = 36 and I will add the two together 1332.0597

You can notice how we do not use the product rule since it has addition there on the bottom.0608

Let us try a few more examples and see if we can recognize our product rule in action.0619

I have thrown in a few more variables this time.0623

The first one is n × n4.0625

It is tempting to say hey maybe this one is just the n4, but if you see missing powers like on that first one, assume that they are 1. 0629

That way you actually have something to add.0638

This would be n1+ 4 and now that they actually have your exponents added together, we will get n5.0640

Now you know if you strictly look at the product rule it only talks about putting two things together, you can even apply it for more than two things.0655

In this one, I have z2 × z5 × z6 and we can apply the product rule as long as we do it two at a time.0664

Let us just take the first two z’s here. 0673

This would be z^(2+5), we will get to that other z in a bit.0677

I can see that I have a single z^(2+5) and z6, that simplifies into z7.0686

I can put these two together z^(7+6) and that would be z13.0696

We could have even taken a much bigger shortcut if we simply just added all 3 of them together.0707

You can definitely do this as long as all of their bases are the same, and you are dealing with multiplication.0716

One more, this one is 42 multiplied by 35.0723

We are definitely dealing with multiplication but again, this one has a little bit of a problem.0730

Notice how this one, the bases are not the same. 0737

The condition for using that product rule is that we need the base is the same 0748

and we need multiplication and so we simply can not use the product rule here.0752

If we are going to go any farther with this one, we have to evaluate these exponents.0756

42 would be 16 and then I will go ahead and multiply that by 35 = 243 and then multiply the two together, so 3888.0762

Be very careful that the conditions of the product rule are met before you try and use it.0787

Another good rule that we can use for combining things together is known as the power rule.0797

The way the power rule works is, we have something raised to an exponent and then we go ahead and re-raise all of that to another exponent.0804

It is like we have exponents of exponents.0812

The way the power rule handle these is we will take both of the exponents and we will actually multiply them together.0816

To convince you that this is the way it should work, we are going to deal with another example that has just numbers in it, 0826

and show you that this is the way you should combine your exponents in this case.0833

We are going to take a look at 43 and all of that is being squared. 0838

Say you knew nothing about the power rule, how could you end up interpreting this?0845

One way is you could use again repeated multiplication to interpret your exponents. 0849

In the first exponent I'm going to interpret is this two right here.0856

I’m going to take my base, 43 and it will be multiplied by itself twice, you can see I have two of them.0860

I want to go further with this, now I will interpret both of these 3s as 4 being multiplied by itself 3 times.0873

Now that I have expanded it out entirely, I’m going to work to go ahead and package it all up.0891

What I can see here is that I have a whole bunch of 2s and they are all going to be multiplied together.0897

In fact I have a total of 6, this is 46.0905

Go ahead and observe exactly what happened with the exponents.0913

Originally we started with 3 and 2 and if you multiply those together like the power rule says we should sure enough, you get 6.0917

If you fall along the process you can see why that happens. 0925

Now that we know more about this rule, let us move on.0929

The power rule which we learned earlier can be applied to multiplication and division.0937

The way it does it is we can take two things that are being raised to a power and then that giving each of them that exponent.0942

When dealing with division, if we have things being divided, say, a ÷ b we can take that exponent and give it to each of them.0953

What I’m trying to say here is that if you have an exponential expression and it is raised to a power,0964

you are going to apply it to all the parts on the inside as long as those parts are being multiplied or divided.0968

This one helps us when we get to those larger expressions.0973

Let us try a few of these up, we simply want to use the power rule for each expression rated out.0981

The first one is 62 and all of that is being raised the fifth power.0988

I'm not going to expand that out, I’m just going to go directly to the rule.0993

This would be 6 then we have 2 × 5.0998

Taking care of just the exponents this would be 610.1004

I could continue to try and figure out what number that is but I’m only interested in considering what the power rule is.1010

Let us try another one.1017

This is z4and all of that is being raised to the 5th.1019

I can take my exponents and I will multiply them together. 1026

This will be z20.1031

Let us get into a few that are a little bit more complicated and see how we can tackle this.1036

In this next one I have 3 × a2 × b4.1042

One of the first things I need use is my power rule and give that 5 to all of the parts.1047

Let us write that to each of these.1057

I’m going to do the 53, I'm going to give it to the a2 and I'm going to give it to the b4.1061

It must go on to all of those parts and it is okay since all these parts were multiplied in the original.1070

Now that I have this, I will go ahead and start to simplify each of these.1077

My 35 is 243, then I will multiply these exponents together and get a10.1086

We will multiply the 4 and 5 together and would give us b20.1097

This one is a completely condensed down far as I can go.1104

Let us see one that involves division.1109

I'm going to give the 3 to the top and bottom.1116

The top will be 33 and the bottom will be x3.1121

This will give us 27 all over x3.1133

Let us try one more 4 + x2.1140

What does the power rule say we should do with this one?1145

I threw this one in there because you should not use the power rule on this one. 1149

Why not? Why can not we use the power rule here?1156

Notice the parts on the inside, they are being added 4 + x instead of being multiplied.1159

Since it is being added instead of multiplied, I can not simply give the 2 to all the parts.1170

This situation down here is completely different from this one because of that addition.1175

How could I take care of this?1181

What we will see in one of our future lessons is that we will handle this one using repeated multiplication1185

and we simply have to multiply those two terms together.1193

For now I want to point out is you should not use the power rule on that situation simply does not work.1196

All of these rules are extremely important and you should be very comfortable with them.1207

I recommend memorizing all of them.1212

Usually a lot of practice with them will get you comfortable with recognizing when you should and should not use them. 1214

In addition, you should be comfortable with these that you can use more than one of them in a single problem.1223

Maybe you will end up using the power rule and then end up using the product rule all in the same one to condense and simplify an expression.1229

That is how comfortable with these rules you should be. 1237

We can get a better handle and the least a little bit more comfortable with these we will give it a try.1243

We will try and use many of these rules together to simplify the following expressions.1248

Okay, this first one is (5k3 / 3)2.1253

One of the very first rules that I'm going to use, is I'm going give that 2 to the numerator and to the denominator of that fraction.1262

You got to be careful in doing this, the top is 5k3 and the bottom is just 3.1272

We will give 2 to each of those.1280

Now that I have taken care of that rule I can see I have a little bit more of work to do on top. 1285

I have multiple parts in there all being multiplied and I’m going to give the 2 to each of those parts. 1289

Here is my 5k3 so we will give the 2 to the 5 and we will give the 2 to the k3.1296

Now that we spread out our variable amongst all of the parts that are being multiplied and divided, let us see if we can continue.1308

52 that would be 25 and now I have k3 and all of that is being raised to the power of 2.1316

What rules should we use for that?1326

As we only have a single base in there, this is that situation where we multiply the two together so this will be k6.1330

Now I simply have 9 on the bottom from 32.1341

I do not see anything else that I can combine or any other rules that I can use, this is good to go.1346

Be careful in using multiple rules.1353

Let us take a peek at the second one here. 1357

This one is (-3 × x × y2)3.1359

In addition that is being multiplied by (x2 × y)4, lots and lots of different exponents flying around.1366

Let us recognize that we do have some multiplication on the insides of those parentheses, 1376

I will be able to spread out that exponent amongst all of its parts.1381

What parts do I have in there?1386

I have -3, x, y2 so we will put 3 on each of those.1388

Let us do the same thing for the other one.1403

I have x2 and y4, both of these looks like they need 4.1406

(x2)4 y4, not bad.1413

Now that I have applied that rule, let us go through and start cleaning other things up.1421

Starting at the very beginning I have -33.1427

That is -3 × -3 × -3 = -27.1431

Right now I have x3 just as it is.1440

It looks like this next part that is a good place where I can multiply the exponents together, y6.1445

Here is a couple of more situations where again I can just multiply those exponents together x8 y16.1456

It is important not to stop there because all of these pieces are still being multiplied together.1470

I can see that I have a couple of x and a couple of y.1476

Let us use our commutative property for multiplication to change the order of these parts.1481

x3 x8 y6 y16.1488

The reason why I’m doing this is so we can better see that I need to apply one more of our rules that will be the product rule.1494

Both of these have the same base and I can simply add their exponents together, the 3 and 8.1502

-27 x^(3+8) is 11 and we can do the same thing down here with our y since they have the same base as well and are being multiplied.1512

y^(6+16) would be 22.1531

We finally got down to our answer and apply that to all the rules that we could.1538

You can be very comfortable with mixing and matching and putting these rules together.1543

Just be very careful that you do them correctly.1548

A very important rule that we will need is the 0 exponent rule.1555

What this rule says is that when you raise anything to the 0 power like a over here, that what you get is simply 1.1561

This is probably one of the most curious rules that you will come across.1570

After all, usually when you are dealing with 0, you are familiar with multiplying by 0 and getting 0 or even adding 0 and do not change anything.1573

Why is it that when we take something to the 0 power, it becomes 1?1581

One reason that we do this is if we want to be consistent with the rest of our rules. 1587

After all, we have been building up a lot of other rules, combining things, our repeated multiplication, we wanted to the mesh well with those other rules.1593

Let us see how it actually does need to be defined as 1, so that it fits with everything else we are trying to do.1600

I’m going to look at 60 × 62 and we will do this twice. 1609

The first time that I go through this, I'm going to end up using my product rule.1617

The reason why I’m doing this is that I have the 6 and the 6 are both the same base and that rule says I need to add the exponents together. 1622

This would be 6^(0+2) now the addition is not so bad 0+2 is 2 so this would end up being 36.1631

For this rule to stay consistent with that, I need it to also equal 36.1648

You can see that if I look at just 62 by itself, this guy over here that does equal 36.1656

What should I call this thing to multiply by 36 so that my final result is 36.1666

You should not have to think on it to long, there is only one thing that I can call 60.1676

I must call it a 1 so that when it is multiplied by that 36, it still is 36 as the final result. 1683

It is the only way that we can define something to the 0 power so that it mixes well with all the rest of our properties.1691

Let us play around this rule a little bit.1701

This new rule is the quotient rule.1707

In this one we are dealing with two things that have the same base so a and a and they are being divided.1710

When we divide like this, it looks like we can simply take their exponents and subtract them.1720

It is a lot like our product rule only that one dealt with multiplication and addition, this one has division and subtraction. 1726

Some important things to know of course we must have the same base for this to work out.1735

We are dealing with division and subtraction.1740

This is a quick example to see why this might work.1743

Let us look at 54/ 52.1747

Suppose I knew nothing about the quotient rule, one way that I could just end up dealing with this, is looking at it as repeated multiplication 5 × 5 × 5.1754

Then I could look at the bottom and say okay what is 52?1768

That would be 5 × 5 and then I can go through canceling out my extra 5 in the top and in the bottom.1774

This would mean 5 × 5, which is the same as 52. 1784

If we look at the result of this, you know what happened to the original versus the answer in the end, 1792

you can see that we also have to is subtract those exponents.1798

4-2 does equal 2.1801

This is the way that we will handle things that have the same base and we are dealing with division.1804

We will simply subtract their exponents. 1810

Now, unfortunately this does lead to a very interesting situation and potentially it might give us some negative exponents.1816

To see why this might happen, let us use an example with numbers so 1825

we get a good sense of some of the strange things that we want to be prepared for.1829

This one I have 24 ÷ 25.1833

The first way I’m going to handle this, is I am going to use my rule for the quotient rule.1838

This will be 2^(4-5).1845

If you take 4 – 5, you will get 2-1.1852

Now I can see that is what the product rule tells me to do, but you know what exactly does that mean to have 2-1.1858

After all, when we are dealing with just whole numbers, then I would often look at this as repeated multiplication, but how can I multiply 2 by itself?1867

A -1 number of times and that just does not seem to make sense. 1875

To get a handle of what that means, I'm going to look at the original problem as repeated multiplication.1879

This would be 2 × 2 × 2 × 2 all over 2 × 2 × 2 × 2 × 2.1886

I have 4 of them on top and 5 of them on the bottom.1898

Let us go through and cancel out all of our extra 2s, 4 from the top and 4 from the bottom.1902

You can see from doing this, there is only one left and it is on the bottom so this would be ½ .1911

Now comes the important part that if we look at each sides of our work out, we have used valid rules and we have done things correctly.1918

What I have here are two different expressions which are the same thing.1927

This gives us a good clue on what we should do with those negative exponents. 1934

We want to interpret our negative exponents by putting the base in the denominator of our fraction.1938

It is how we will end up handling these things.1945

It will take a little bit more work to get comfortable with these, let us see what else we can do.1949

The rule that we have just developed is anytime we have a base raised to a negative exponent we will put it in the denominator of the fraction1960

and we will change that base, a rule change that exponent to a positive number. 1969

Another way to say that is if you have an expression raised to a negative power it can be rewritten in the denominator with a positive power. 1975

It leads to something very interesting. 1983

If you have a negative in the top of your fraction and a negative in the bottom, 1986

then what you can end up doing is changing the location of where those things end up.1991

I have a^-n and it was on the top but now it is in the bottom. 1997

I had b^-n to in the bottom, now it is in the top.2003

A good rule of thumb that I give my students to keep track of what to do with that negative in the exponent 2010

is think of it as changing the location of where something is.2016

If the negative exponent is in the numerator, think of the top, go ahead and move it to the bottom at your denominator and make it positive. 2020

The other situation, if it is already in the bottom at your denominator, then move it to the numerator the top and make it positive.2030

It will help you handle these in a nice quick way.2040

Now that we know a lot more about our exponents and especially those negative ones, let us get into using them a bit more.2047

Here I have ¼-3.2055

One of the first rules that I'm going to use is to spread out that -3 onto the top and bottom of my fraction.2062

I have 1-3 / 4-3.2070

Here is where I’m going to handle that negative exponent.2077

That 1-3 on the top, I’m going to move it to the bottom and make its exponent positive.2080

I’m going to do the same thing with a 4.2090

4-3 now it will go to the top and its exponent will now be positive as well.2091

From here, I just go through simplify it, 43 = 64, 13 = 1 so it is 64.2098

Let us try the same thing with the next one and notice how things are changing location.2116

2-3 will end up in the bottom as 23.2123

3-4 that one is going to go into the top as 34.2130

I will go ahead and simplify from here.2139

There may be a few situations where some things will have positive exponents and some things will have negative exponents. 2144

The only ones that will change locations will be the ones with negative exponents.2155

If I have an example like x2 and y-3, then the x2 will remain in the same spot but the y must go into the bottom since it had the negative exponent.2160

Watch out for those.2172

Let us see if we can definitely combine many more of our rules together and simplify each of these expressions.2177

We got lots of rules to keep track of so we will just do this carefully, bit by bit. 2184

In this first one I have the (x2 / 2y3)-3.2191

One of the biggest features I can see here is probably that fraction.2198

I’m going to give the -3 to the top and the bottom of my fraction.2204

I have x2 and will give it -3 and I will do the same thing with the bottom. 2209

Now one thing I can see on the top is I have x2 and that is being in turn raised to another power.2218

Our rule for that says we need to multiply the exponents x-6.2225

What to do with the bottom?2233

In the bottom we have some multiplication in there, so 2 × y3.2234

I need to give that -3 to each of the pieces down here.2239

Moving on, bit by bit, I will deal with that negative on the x and eventually that negative on the 2.2250

Let us see what we need to do with the y3 and y-3.2258

The rule says I need to multiply those things together and get -9.2262

I think I have used all of my product rules and power rules and quotient rule.2269

It is time to use that negative exponent rule.2273

Anything that has a negative exponent on it is on the top and I’m going to put in the bottom.2277

If it was on the bottom, it is going right to the top.2281

In the bottom, x6 and that guy is done.2287

In the top, 23 and y9.2292

One last thing to go ahead and clean this up, 23 is 8.2298

I will just ahead and put it in there.2303

Our final simplified expression for this one is 8y9 / x6.2307

It is quite a bit of work, but it is what happens.2314

Let us do another one. 2318

This one is for 4h-5 / m – (2 × k).2320

I do not see a whole lot of rules in terms of spreading things out over multiplication, but one thing I do want to do is take care of those negative exponents.2327

I have one of them up here on the h and another one here on the m.2336

Let me first write down the things that do not have negative exponents, they will be in exactly the same spot. 2341

The h-5 I need to put that in the bottom as h5.2348

The m-2 it now needs to go to the top as m2. 2354

Okay, I would continue simplifying from here if I could but I think I do not see anything else that has the same base.2360

I will leave it as it is and will call this one done.2368

It is time to put all of our rules in practice and see if we can do a much harder problem.2376

This one is (39 × x2 × y)-2 / 33 × x-4 × y.2381

One thing I want to point out with this one is that when you are applying the rules, you do have a little bit of freedom on which ones you do first. 2395

Experiment with trying the rules in a different order and see if you come up with the same answer.2403

As long as you apply the rules carefully and correctly, it should work out just fine.2408

What shall we do with this one?2414

I'm going to go ahead and take that -2 and spread it out on my x2 and y since both of those are being multiplied on the inside there.2417

39 now have an (x2)-2.2425

I also have a y-2 as well.2434

I will put that -2 in both of them.2438

Let us go ahead and multiply our exponents here, 39 then 2 × -2 would be x-4 y-2.2448

Let us see where can I go from here. 2471

I like dealing with those negative exponents and changing the location of things so I do not have to worry about negatives.2472

Let us do that first. 2479

Everywhere I see a negative exponent that part will change its location.2481

I’m going to leave the 39 for now and leave my 33 and let us take care of this one.2486

x-4 will be x4 in the bottom.2493

This one which is x-4, let us put that in the top.2499

y-2 is in the bottom and this y already has a positive exponent, so no need to change that one.2505

Continuing on, let us see what else we can do.2516

These 3s out here, they have exactly the same base so I can subtract their exponents, 9 – 3.2519

I can do the exact same thing with these x’s, subtract their exponents.2529

What should I do with those y’s on the bottom?2541

They have the same base, so I will add those exponents together.2543

Good way to start crunching things down.2550

This will be 36 x0 / y3.2554

36 = 729 x0, remember that is one of our special ones is 1, all over y3.2564

This one crunches down quite a bit, but in the end we are left with 729 / y3.2581

We do not have anything else to combine and so we will consider this one simplified.2589

Only one more to do this one, we want to make an expression that represents the area of the figure.2597

I want something that is a little bit more like a word problem, at least something that we have to drive out of.2604

We will simply use our rules along the way to crunch this down.2609

When I look at the area of a rectangle like this, it is formed by taking the width of that rectangle and multiplying it by the length.2613

I’m not sure what is the width and length here, since both of them are written as expressions.2625

I will write those in, area is the width 4x2 and the length is 8x4.2630

That means I can take this expression for the area and put it together using some of my rules.2640

Let us rearrange that.2646

I’m going to put my numbers 4 and 8 together, and my variables and exponents together. 2648

4 × 8 = 32 and I have x2 and x4.2657

I can do these by adding them together since they have the same base giving me x6.2665

This expression would be equal to the area of the rectangle.2671

It has quite a lot of rules to enjoy but again with a little bit of extra practice, they should become a little more familiar.2677

Watch for those special ones like raising something to the 0 power so that you know that is always 1.2683

Thank you for watching