Video Transcripts: Exponents

Intro to Exponents 

You already know that we can view multiplication as repeated addition. So, if we had 2 times  3 (2 × 3), we could literally view this as 3 2's being added together. So it could be 2 + 2 + 2.  Notice this is [COUNTING: 1, 2] 3 2's. And when you add those 2's together, you get 6. What  we're going to introduce you to in this video is the idea of repeated multiplication – a new  operation that really can be viewed as repeated multiplication. And that's the operation of  taking an 'exponent.' And it sounds very fancy. But we'll see with a few examples that it's not  too bad. So now, let's take the idea of 2 to the 3rd power (2^3) – which is how we would say  this. (So let me write this down in the appropriate colors.) So 2 to the 3rd power. (2^3.) So you might be tempted to say, "Hey, maybe this is 2 × 3, which would be 6." But remember, I just  said this is repeated multiplication. So if I have 2 to the 3rd power, (2^3), this literally means  multiplying 3 2's together. So this would be equal to, not 2 + 2 + 2, but 2 × ... (And I’ll use a  little dot to signify multiplication.) ... 2 × 2 × 2. Well, what's 2 × 2 × 2? Well that is equal to 8.  (2 × 2 × 2 = 8.) So 2 to the 3rd power is equal to 8. (2^3 = 8.) Let's try a few more examples  here. What is 3 to the 2nd power (3^2) going to be equal to? And I'll let you think about that for  a second. I encourage you to pause the video. So let's think it through. This literally means  multiplying 2 3's. So let's multiply 3 – (Let me do that in yellow.) Let's multiply 3 × 3. So this is  going to be equal to 9. Let’s do a few more examples. What is, say, 5 to the – let's say – 5 to  the 4th power (5^4)? And what you'll see here is this number is going to get large very, very,  very fast. So 5 to the 4th power (5^4) is going to be equal to multiplying 4 5's together. So 5^4  = 5 × 5 × 5 × 5. Notice, we have [COUNTING: 1, 2, 3] 4 5's. And we are multiplying them. We  are not adding them. This is not 5 × 4. This is not 20. This is 5 × 5 × 5 × 5. So what is this  going to be? Well 5 × 5 is 25. (5 × 5 = 25.) 25 × 5 is 125. (25 × 5 = 125.) 125 × 5 is 625. (125  × 5 = 625.) 

The Zeroth Power 

If we think about something like two to the third power, we could view this as taking three twos and multiplying them together, so two times two times two, or equivalently, we could say this  is the same thing as taking a one and then multiplying it by two three times, so actually, let's  just go with this definition right over here, and this, of course, is going to be equal to eight.  Now what would, based on this definition I just did, what would two to the second power be?  Well, this would be one times two twice, so one times two times two, which, of course, would  be equal to four. What would two to the first power be? Well, that would be one, and we would multiply it by one two, one times two, which, of course, is equal to two. Now let's ask ourself  an interesting question. Based on this definition of what an exponent is, what would two to the zeroth power be? I encourage you to just think about that a little bit. If you were the mathematics community, how would you define two to the zero power so it is consistent with  everything that we just saw. Well, the way we just talked about it, we just said exponentiation  is you start with a one and you multiply it by the base zero times, so we're not gonna multiply  it by any two, so we're just gonna be left with a one. So does this make sense that two to the  zero power is equal to one? Well, let's think about it another way, and let's do a different base. That was with two, but let's say we have three and I could say three to the fourth power, that's three times three times three times three which is going to be equal to 81, and let me just  write down that this is going to be equal to 81. If I said three to the third power, that's three  times three times three which is 27. Three to the second power is equal to nine. Three to the  first power is equal to three. Do you notice a pattern every time we decrease the exponent  here by one? We want three to the fourth, and now we go three to the third. What happened  to the value? Well, going from 81 to 27, we divided by three, and that makes sense, because  we're multiplying by one less three, so we divide by three to go from 81 to 27, we divide by 

three again if our exponent goes down by one, and we divide by three again when we go from nine to three, divide by three, so based on this, what do you think three to the zero power  should be? Well, the pattern is every time we decrease our exponent by one, we divide by the base, and so we should divide by three again would be the logic if we follow that pattern, and  so three divided by three would get us one again. So I know it might seem a little bit counter  intuitive that something to the zeroth power is going to be equal to one, but this is how the  mathematics community has defined it because it actually makes a lot of sense. Either if you  view an exponent as taking a one and multiplying it by the base the exponent number of  times, so I'm gonna multiply one by two three times, or if you just follow this pattern, every  time you decrease the exponent by one, you're going to be dividing by the base. Either of  those would get you to the conclusion that two to the zero power is one, or three to the zero  power is one, or frankly, any number to the zero power is one. So if I have any number, let's  say I have some number a to the zero power, this is going to be equal to one. Now I have an  interesting question for you, and let's just say this is the case when a does not equal zero. I'll  leave you a little bit of a puzzle for you to think about. What do you think is zero to the zeroth  power? What should zero to the zero power be? And what's interesting about zero to the zero power is you'll get a different answer if you use this technique versus if you use this technique right over here. This technique would actually get you to being one, while this technique  would have you divide by zero, which we don't know how to do. Anyway, I'll leave you there to ponder the mysteries of zero to the zeroth power. 

Comparing Exponents 

So we are asked to order the expressions from least to greatest and this is from the exercises on Khan Academy and if we're doing it on Kahn Academy, we would drag these little tiles around from least to greatest, least on the left, greatest on the right. I can't drag it around 'cause this is just a picture, so I'm gonna evaluate each of these, and then I'm gonna rewrite them from least to greatest. So let's start with two to the third minus two to the first. What is that going to be? Two to the third minus two to the first. And if you feel really confident, just pause this video and try to figure out the whole thing. Order them from least to greatest. Well two to the third, that is two times two times two, and then two to the first, well that's just two. So two times two is four, times two is eight, minus two, this is going to be equal to six. So this expression right over here could be evaluated as being equal to six. Now, what about this  right over here? What is this equal to? Well let's see, we have two squared plus three to the  zero. Two squared is two times two and anything to the zero power is going to be equal to  one. It's an interesting thing to think about what zero to the zeroth power should be but that'll  be a topic for another video. But here we have three to the zeroth power, which is clearly  equal to one. And so we have two times two plus one. This is four plus one, which is equal to  five. So the second tile is equal to five. And then three squared, well three squared, that's just  three times three. Three times three is equal to nine. So if I were to order them from least to  greatest, the smallest of these is two squared plus three to the zeroth power. That one is  equal to five, so I'd put that on the left. Then we have this thing that's equal to six, two to the  third power minus two to the first power. And then the largest value here is three squared. So  we would put that tile, three squared. We will put that tile on the right, and we're done. 

Exponents of decimals 

What we're going to do in this video is get some practice evaluating exponents of decimals.  So let's say that I have 0.2 to the third power. Pause this video, see if you can figure out what  that is going to be. Well, this would just mean that if I take something to the third power, that means I take three of that number and I multiply them together. So it's 0.2 times 0.2 times 0.2.

Well, what is this going to be equal to? Well, if I take 0.2 times 0.2, that is going to be 0.04.  One way to think about it, two times two is four and then I have one number behind the  decimal to the right of the decimal here. I have another digit to the right of the decimal right  over here, so my product is going to have two digits to the right of the decimal, so it'd be 0.04. And then if I were to multiply that times 0.2, so if I were to multiply that together what is that  going to be equal to? Well, four times two is equal to eight and now I have one, two, three  numbers to the right of the decimal point, so my product is going to have one, two, three  numbers to the right of the decimal point. So now that we've had a little bit of practice with  that, let's do another example. So let's say that I were to ask you, what is 0.9 squared? Pause this video and see if you can figure that out. All right, well this is just going to be 0.9 times 0.9.  And what's that going to be equal to? Well, you could just say nine times nine is going to be  equal to 81, and so, let's see, in the two numbers that I'm multiplying I have a total of one, two numbers, or two digits, to the right of the decimal point so my answer's going to have one, two digits to the right of the decimal point. So put the decimal right over there and I'll put the zero,  so 0.81. Another way to think about it is nine-tenths of nine-tenths is 81 hundredths, but there  you go. Using exponents, or taking exponents of decimals is the same as when we're taking it of integers. It's just in this case you just have to do a little bit of decimal multiplication. 

Powers of fractions 

Let's go through more exponent examples. So to warm up, let's think about taking a fraction  to some power. So let's say I have 2/3, and I want to raise it to the third power here. Now,  we've already learned there are two ways of thinking about this. One way is to say let's take  three 2/3's. So that's one 2/3, two 2/3's, and three 2/3's. So that's one, two, three, 2/3. And  then we multiply them. And we will get-- let's see, the numerator will be 2 times 2 times 2,  which is 8. And the denominator's going to be 3 times 3 times 3 times 3, which is equal to 27.  Now, the other way of viewing this is you start with a 1, and you multiply it by 2/3 three times.  So you multiply by 2/3 once, twice, three times. You will get the exact same result here. So  let's do one more example like that. So lets say I had 4/9, and I want to square it. When I  raise something to the second power, people often say, you're squaring it. Also, raising  something to the third power, people sometimes say, you're cubing it. But let's raise 4/9 to the  second power. Let's square it. And I encourage you to pause the video and work this out  yourself. Well, once again, you could view this as taking two 4/9's and multiplying them. Or  you could view this as starting with a 1, and multiplying it by 4/9 two times. Either way, your  numerator is going to be 4 times 4, which is 16. And your denominator is going to be 9 times  9, which is equal to 81.