Hi, my name is Anthony Varela. And today I'd like to talk about negative exponents. So we're going to look at patterns in negative exponents. And then we're going to practice writing expressions that have negative exponents.
So, first, I'm going to start with a positive exponent. I have 2 squared. And I can think of 2 squared as 2 times 2, using that base number 2 in a chain of multiplication two times, like our exponent tells us. Well, I'm going to decrease that exponent by 1, and how can I write 2 raised to the first power?
Well, I can think of this as just 2, using our base number and a chain of multiplication just one time. But I'd also like to think of this as 2 times 2, which is 2 squared, divided by 2. So I am canceling one of those factors of 2 through this division.
So now let's decrease that exponent again. So I have 2 raised to the power of 0. Well, I know from my previous example I can express this as 2 to the first or 2 divided by 2, canceling one of those factors through division. Well, 2 divided by 2 is 1. And we know that anything raised to the power of 0 equals 1.
So let's decrease this exponent again. And now we're getting into our negative exponents. So how can I think of 2 raised to the power of negative 1? Well, based on my patterns here, I know I can write this as 1 divided by 2. So 2 raised to the negative first power is 1 divided by 2.
So now we see that when I decrease the exponent again, I'm dividing by 2. So 1/2 divided by 2 is 1/4. Let's decreased that exponent again. And this is going to be 1/4 divided by 2, which is 1/8.
So let's take a closer look at our negative exponent pattern. Well, 1/2 can be thought of as 1 over 2 to the first power. 2 to the first power is 2. 4 is 2 to the second power. So I can write 1/4 as 1 over 2 squared. And I can write 1/8 as 1 over 2 cubed.
And now look at our original expression and our fraction here. This negative 1 is now a positive 1 in a denominator. This negative 2 is now a positive 2 in a denominator. And this negative 3 is now a positive 3 in a denominator.
So what we can say then is that a base number raised to a negative exponent is the same as 1 over that base to a positive exponent. And we can also say that 1 over a base number to a negative exponent equals that base number to the positive exponent. So we're going to write that down. Negative exponents can be rewritten to have positive exponents. But notice that's also involving a fraction.
And our other rule, as well, where we have a negative exponent in the denominator of a fraction, can be rewritten is a positive exponent in a numerator. And notice, we don't have to write a denominator at all because it would be over 1. We don't need to write that.
All right. So now let's practice then taking expressions that have negative exponents and we're going to rewrite them so they have positive exponents. So here we have 5 raised to the negative fourth power. So we're going to be using this equation up here. So we're going to create a fraction. 1 is our numerator. And denominator is our expression, but just with a positive exponent instead, so 5 to the fourth power.
Pretty simple. Let's try another one. 8 raised to the negative third power. This is going to equal a fraction where 1 is our numerator. And our denominator has our base raised to that positive power. So 8 raised to the positive third.
All right. Now let's go the other way. So we're going to start with our fraction here. And then we're going to write it using a negative exponent. So here we have 1 over 9 squared. Well, I'm going to rewrite this as 9 to the negative second power. So we're getting rid of this fraction, or really we're reporting our denominator as 1. And we have 9 raised to a negative power instead of a positive power.
One more example. We have 1 over 3 to the sixth power. We're going to write this using a negative exponent. This is our base 3 raised to the negative sixth power.
And lastly, we're going to apply this equation here. So we're going to have a fraction. And in our denominator, we'll have a negative exponent. And we're going to rewrite this. So we have 1 over 4 to the negative seventh. This would be simply 4 to the seventh power. So we went from a negative exponent to a positive exponent.
In our last example, 1 over 6 to the negative 6. We're going to rewrite this with no fraction and a positive exponent. This is 6 to the sixth power.
So let's review what we talked about today. Negative exponents can be written with positive exponents. So if you ever encounter a base number raised to the negative power, if you'd like, you can rewrite this as 1 over that base to the positive power.
And if you ever see 1 over a base to a negative power, you can rewrite this as that base to the positive power.
Well, thanks for watching this video on negative exponents. Hope to catch you next time.