Hi, and welcome. My name is Anthony Varela. And today, I'm going to talk about exponents. So we're going to answer your questions, what are exponents? When will I see exponents? And are there any special exponents I need to know about?
So to introduce this idea of exponents, we're going to take what we already know about multiplication. So multiplication is repeated addition. So we can take a number, 6. And if we add to that, 6 plus 6 plus 6, we can rewrite this as a multiplication expression-- 6 times 4.
Now, both of these equals 24. So let's take a look at our expression, 6 times 4. 6 was the addend in our original chain of addition. And 4 told us how many times we add that number. Now, what does this have to do with exponents?
Well, an exponent is repeated multiplication. So we can take a number, such as 3. If you multiply that by 3, multiply that by 3 again, multiply that by 3 again, we could write this using an exponent. We can say 3 raised to the fourth power. So let's take a look at how we wrote this-- 3 raised to the fourth power. There's that big 3, and then a smaller 4, which is written higher up than the 3.
Well, the 3 is what we call the base. That's the number that we're going to use in a chain of multiplication. And the 4 tells us the number of times we multiply that base. And that's actually what the exponent is. So we're going to write this down, that an exponent is repeated multiplication. And in general, we can write this as b to the power of n, where b is that base number and n is the exponent, or the power.
So how do we read exponents? So you've already heard me say 3 to the fourth power. Another way you might hear this spoken is 3 to the power of 4, or 3 raised to the fourth. Those are all perfectly fine ways to read that.
So let's try another one. We have 7 as the base and 8 as the exponent. You might hear someone say 7 to the 8. You might hear 7 raised to 8, 7 raised to the power of 8-- those are all different ways to read this. So generally, we say that the base is raised to an exponent power.
So what are some common exponents? I'm going to talk about the exponents of 2 and 3. So when we see a base raised to the power of 2-- so here we have 3 to the power of 2.
Most commonly, you will hear this spoken as 3 squared. And the reason why we say 3 squared is because we can think of the shape of a square. All sides are equal, so we express the area of this square as s squared, where s is that side length. So the exponent 2 is associated with being squared.
So the exponent of 3, now, is commonly referred to as cubed. So 3 to the power of 3 is 3 cubed. And we can think of the shape of a cube. It has 3 dimensions, and all side lengths are the same, so we can express the volume as the side length cubed. So let's write that down, that common exponents 2 and 3 are referred to squared when we're talking with exponent of 2 and cubed when we're talking about the exponent of 3.
So now, how about some special exponents? I'd like to talk about two more special exponents-- 0 and 1. I'm going to talk about the exponent of 1 first. So what if the exponent is 1? Well, let's take this expression-- 5 to the first power.
Now, remember from before that the exponent told me how many times to use that base in a chain of multiplication. So my exponent of 1 tells me to use 5 just once. So 5 to the first power equals 5.
Now, we can extend this property to variables. k to the first power is simply k, using k as a factor in chain in a chain of multiplication just one time. I can express this using an entire expression, too-- k plus 2. All of that raised to the power of 1 means I'm going to be using k plus 2 in a chain of multiplication just one time. So k plus 2 raised to the power of 1 equals k plus 2.
So what's the pattern here? Anything to the power of 1 remains the same. It's just that base. So we're going to write that down, that a to the power of equals a.
Now, what if the exponent is 0? So let's see. Let's take this expression, 5 to the third. Now, I know that this equals 5 times 5 times 5, using 5 in a chain of multiplication three times. So now, I'm going to decrease this exponent by 1 and say 5 squared.
Well, I know that this is using 5 in a chain of multiplication two times, but I'm going to write this as 5 cubed over 5. Notice that through this division by 5, I'm taking away one of those factors. 5 times 5 times 5 divided by 5 equals 5 times 5.
Well, what does this have to do with an exponent of 0? Let's keep on getting closer to that exponent of 0. 5 to the first power-- well, I already know that this equals 5. That was from our last slide. But I'm going to write this as 5 squared divided by 5, showing again that I can take away one of those factors through this division.
So now, I'm getting to my exponent of 0. How can we evaluate 5 raised to the power of 0? A common mistake is that we think that just equals 0, because that's what we think about with multiplication. Anything times 0 equals 0. But I'm going to write this as 5 to the first power over 5.
Following this pattern, I know I can do that. And 5 to the first power equals 5. So I can write this as 5 over 5 over 1. So anything to the power of 0 equals 1. So we're going to write that down, that a to the power of 0 equals 1.
So let's review our notes. Today, we talked about an exponent being repeated multiplication. We talked about the general way to write this as b to the power of n, where b is a base number and n is the exponent. And we generally say that the base number is raised to an exponent power. Then we talked about some common exponents, 2 and 3, where we say that the power of 2 is that quantity squared. And with the power of 3, it's that quantity cubed.
Then we talked about some special exponents. Anything raised to the power of 1 equals that base. And anything raised to the power of 0 just equals 1, no matter what that base number is. So thanks for watching this introduction to exponents. I hope to catch you next time.