### 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

^{3}b

^{5})(6a

^{2}b

^{2}c

^{4})

- 12a
^{3 + 2}b^{5 + 2}c^{4}

^{5}b

^{7}c

^{4}

^{7}y

^{3})(9x

^{6}yz

^{2})

- 36x
^{7 + 6}y^{3 + 1}z

^{13}y

^{4}z

^{3})(6m

^{2}n

^{7})

- 30m
^{1 + 2}n^{3 + 7}

^{3}n

^{10}

^{4}b

^{9}c

^{2})

^{3}

- (4)
^{3}(a^{4})^{3}(b^{9})^{3}(c^{2})^{3} - 64a
^{4 ×3}b^{9 ×3}c^{2 ×3}

^{12}b

^{27}c

^{6}

^{2}y

^{3}z

^{4})

^{2}

- (8)
^{2}(x^{2})^{2}(y^{3})^{2}(z^{4})^{2} - 64x
^{2 ×2}y^{3 ×2}z^{4 ×2}

^{4}y

^{6}z

^{8}

^{3}h

^{6}i)

^{4}

- (5)
^{4}(g^{3})^{4}(h^{6})^{4}(i)^{4} - 625g
^{3 ×4}h^{6 ×4}i^{4}

^{12}h

^{24}i

^{4}

^{2}b

^{3}c

^{3})

^{2}

- (7)
^{2}(a^{2})^{2}(b^{3})^{2}(c^{3})^{2} - 49a
^{2 ×2}b^{3 ×2}c^{3 ×2}

^{4}b

^{6}c

^{6}

^{5}h

^{3}i

^{4})

^{2}( − 4g

^{2}h

^{6}i

^{7})

^{3}

- [(3)
^{2}(g^{5})^{2}(h^{3})^{2}(i^{4})^{2}][( − 4)^{3}(g^{2})^{3}(h^{6})^{3}(i^{7})^{3}] - 9g
^{10}h^{6}i^{8}( − 64)g^{6}h^{18}i^{21} - (9)( − 64)(g
^{10}×g^{6})(h^{6}×h^{18})(i^{8}×i^{21}) - − 576(g
^{10 + 6})(h^{6 + 18})(i^{8 + 21})

^{16}h

^{24}i

^{29}

^{5}h

^{6}i

^{7})

^{3}( − 8g

^{2}h

^{3}i

^{4})

^{2}

- [(5)
^{3}(g^{5})^{3}(h^{6})^{3}(i^{7})^{3}][( − 8)^{2}(g^{2})^{2}(h^{3})^{2}(i^{4})^{2}] - 125g
^{15}h^{18}i^{21}(64)g^{4}h^{6}i^{8} - (125)(64)(g
^{15}×g^{4})(h^{18}×h^{6})(i^{21}×i^{8}) - (125)(64)(g
^{15 + 4})(h^{18 + 6})(i^{21 + 8})

^{19}h

^{24}i

^{29}

^{2}y

^{5}z

^{6})

^{2}( [5/6]x

^{4}y

^{7}z

^{3})

^{3}( x

^{5}y

^{4})

^{4}

- ( − [1/3] )
^{2}(x^{2})^{2}(y^{5})^{2}(z^{6})^{2}( [5/6] )^{3}(x^{4})^{3}(y^{7})^{3}(z^{3})^{3}(x^{5})^{4}(y^{4})^{4} - [1/9]( x
^{4}y^{10}z^{12})[125/216]( x^{12}y^{21}z^{9})( x^{20}y^{16}) - [1/9] ×[125/216]( x
^{4}×x^{12}×x^{20})( y^{10}×y^{21}×y^{16})( z^{12}×z^{9})

^{36}y

^{47}z

^{21}

*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.

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

### Algebra 1 Online Course

### Transcription: Integer Exponents

*Welcome back to www.educator.com.*0000

*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 5 ^{3}.*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 3 ^{4} that happens to be negative sign out front and my next example, it is almost the same thing but it is -3^{4}.*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 2 ^{4} power be multiplied by 2^{3}, let us pretend I knew nothing about the product rule.*0342

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

*I can do the same for 2 ^{3} 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 2 ^{7}.*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 -7 ^{5} × -7^{3}.*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, -7 ^{8}.*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 p ^{13} 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 6 ^{4} and see what value that is.*0593

*I will get 1296 then I will get 6 ^{2} = 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 × n ^{4}.*0625

*It is tempting to say hey maybe this one is just the n ^{4}, 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 n ^{5}.*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 z ^{2} × z^{5} × z^{6} 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 z ^{6}, that simplifies into z^{7}.*0686

*I can put these two together z^(7+6) and that would be z ^{13}.*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 4 ^{2} multiplied by 3^{5}.*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

*4 ^{2} would be 16 and then I will go ahead and multiply that by 3^{5} = 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 4 ^{3} 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, 4 ^{3} 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 4 ^{6}.*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 6 ^{2} 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 6 ^{10}.*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 z ^{4}and 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 z ^{20}.*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 × a ^{2} × b^{4}.*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 5 ^{3}, I'm going to give it to the a^{2} and I'm going to give it to the b^{4}.*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 3 ^{5} is 243, then I will multiply these exponents together and get a^{10}.*1086

*We will multiply the 4 and 5 together and would give us b ^{20}.*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 3 ^{3} and the bottom will be x^{3}.*1121

*This will give us 27 all over x ^{3}.*1133

*Let us try one more 4 + x ^{2}.*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 multiplication*1185

*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 (5k ^{3} / 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 5k ^{3} 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 5k ^{3} so we will give the 2 to the 5 and we will give the 2 to the k^{3}.*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

*5 ^{2} that would be 25 and now I have k^{3} 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 k ^{6}.*1330

*Now I simply have 9 on the bottom from 3 ^{2}.*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 × y ^{2})^{3}.*1359

*In addition that is being multiplied by (x ^{2} × 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, y ^{2} so we will put 3 on each of those.*1388

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

*I have x ^{2} and y^{4}, both of these looks like they need 4.*1406

*(x ^{2})^{4} y^{4}, 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 -3 ^{3}.*1427

*That is -3 × -3 × -3 = -27.*1431

*Right now I have x ^{3} just as it is.*1440

*It looks like this next part that is a good place where I can multiply the exponents together, y ^{6}.*1445

*Here is a couple of more situations where again I can just multiply those exponents together x ^{8} y^{16}.*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

*x ^{3} x^{8} y^{6} y^{16}.*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 6 ^{0} × 6^{2} 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 6 ^{2} 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 6 ^{0}.*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 5 ^{4}/ 5^{2}.*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 5 ^{2}?*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 5 ^{2}.*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 2 ^{4} ÷ 2^{5}.*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 fraction*1960

*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, 4 ^{3} = 64, 1^{3} = 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 2^{3}.*2123

*3 ^{-4} that one is going to go into the top as 3^{4}.*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 x ^{2} and y^{-3}, then the x^{2} 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 (x ^{2} / 2y^{3})^{-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 x ^{2} 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 x ^{2} 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 × y ^{3}.*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 y ^{3} 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, x ^{6} and that guy is done.*2287

*In the top, 2 ^{3} and y^{9}.*2292

*One last thing to go ahead and clean this up, 2 ^{3} is 8.*2298

*I will just ahead and put it in there.*2303

*Our final simplified expression for this one is 8y ^{9} / x^{6}.*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 h^{5}.*2348

*The m ^{-2} it now needs to go to the top as m^{2}.*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 (3 ^{9} × x^{2} × y)^{-2} / 3^{3} × 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 x ^{2} and y since both of those are being multiplied on the inside there.*2417

*3 ^{9} now have an (x^{2})^{-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, 3 ^{9} 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 3 ^{9} for now and leave my 3^{3} and let us take care of this one.*2486

*x ^{-4} will be x^{4} 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 3 ^{6} x^{0} / y^{3}.*2554

*3 ^{6} = 729 x^{0}, remember that is one of our special ones is 1, all over y^{3}.*2564

*This one crunches down quite a bit, but in the end we are left with 729 / y ^{3}.*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 4x ^{2} and the length is 8x^{4}.*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 x ^{2} and x^{4}.*2657

*I can do these by adding them together since they have the same base giving me x ^{6}.*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 www.educator.com.*2689

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.