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# Patterns in powers of 10

Sal examines patterns in powers of 10. He also introduces the terms exponent and base. Created by Sal Khan.

Video transcript

So let's think about what
happens when we multiply by 10, and see if we can see
some type of a pattern. So let's just start with 1
and multiply that 1 by 10. And this, of course,
is equal to 10. You knew that already. But let's multiply by 10 again. So let's do 1 times 10. And now let's multiply
by another 10. Let's multiply by another 10. So what's this going to be? This is going to be 1 times
10, which is 10, and then that 10 times 10. So it's literally 10 10s or 100. Now let's multiply by 10 again. So we're going to have 1 times
10 times 10 times 10 times 10. What is this going
to be equal to? Well we already know that
1 times 10 times 10 is 100, and then we're going to
multiply that times 10 again. So we're going to literally
have 10 100s which we know is the exact
same thing as 1,000. So what's the pattern here? Well here we multiplied by
10 once and we have one zero. Here we multiplied by 10
twice and we have two zeroes. Here we multiplied
by 10 three times and we have three
zeroes in our answer. Let's do it again,
although this looks like a pretty clear pattern. Let's multiply by 10 four times. So we'll start with a 1. We'll multiply by one
10, that's our first 10, then our second 10, then
our third 10, and now our fourth 10. Well what's this
going to be equal to? Well it's going to be
this thing, I should say, times 10 or 1,000 times
10, which is 1,000s. So let me write that 10 1,000s. So once again, we
started with a one, we multiplied by 10 four
times, and we got 10,000. One followed by one,
two, three, four zeros. Let me make that clear. We start with a 1. We multiplied by 10 one,
two, three, four times, and our product is one followed
by one, two, three, four zeros. So I think you see
the pattern here. But there might be
something you're craving. You're like, hey
Sal, this is a pain to do this repeated
multiplication. I wish there was some
type of operation that just did it for us. The same way that
back in the day when you were doing
repeated addition where if you were saying 10 plus
10 plus 10 plus 10, you said, hey, this is the same
thing as 4 times 10. There should be another thing
for repeated multiplication. If I have 1 times 10
times 10 times 10 times 10, which is really the same
thing as taking four 10s and multiplying them
together because the 0 doesn't change the value. So 10 times 10
times 10 times 10. You're saying, hey,
there should be some type of shorthand for
this, and you're right. There is some type
of shorthand for this and it's called an exponent. So the same way if you take
four 10s and add them up, that's the same
thing as 4 times 10. If you take four 10s
and take their product, this is the same thing as
10 to the fourth power. 10 to the fourth power
literally means taking four 10s and multiplying them together. Or you could view it
as starting with a 1 and multiplying
by 10 four times. So this is also equal to
10 to the fourth power. This right over here,
we started with a 1. We multiplied by 10
three times, which is the same thing
as taking three 10s and multiplying them together. This is 10 to the third power. This right over
here is 10 squared, or 10 to the second power. And this right over
here, we just have one 10 right over here. We just started with the one
and we multiplied by 10 once. This is 10 to the first power. And just so you know a little
bit of the terminology, here 10 is our base. The base is 10. And this 4 that we
have right over here, this is the exponent. So given that notation,
let's just do one practice. Let's think about what--
I'll do new colors-- what 10 to the sixth power is. And I encourage you to pause
the video and think about it. Well there's two ways
to think about it. You could view this as six 10s--
so 10, 10, 10, 10, 10, and 10 and multiplying them together. So we multiply
them all together. Or you could view it
as starting with a 1 and multiplying it
by 10 six times. But either way, what
are you going to get? Well if you're multiplying
1 by 10 six times, you're going to end up with
1 followed by six zeroes. We already saw this pattern. So it's going to be 1 followed
by one, two, three, four, five, six zeros, or one million. So 10 to the sixth power
is the same thing as 1 being multiplied
by 10 six times. We're taking six 10s and
multiplying them together. And that's going to be equal to
1 followed by six zeros, which is this pattern we
saw or one million.