Hi, welcome back. My name is Anthony Varela, and today, I'd like to talk about the properties of exponents when our exponents are fractions or negative numbers. So we're going to review what fractional and negative exponents are. We'll also review the properties of exponents, and then we'll see what these properties look like when they're exponents are negative or fractional.
So let's quickly review then that fractional exponents can be rewritten as radicals and then vise versa. So here, I have the general rule that a number a raised to the fractional exponent, m over n, can be rewritten as the n-th root of a to the m. So an example would be 4 to the 8/3 power. I can rewrite this then as the cubed root of 4 to the 8.
Now, negative exponents can be rewritten as expressions with positive exponents and then vise versa. So here's our general rule that any number a raised to the negative n can be rewritten as 1 over a to the positive n. So an example would be 4 to the negative thirds. I can rewrite this as 1 over 4 to be positive third.
So now, let's review our properties of exponents, and there are five different properties that I'd like to review with you. The first one is the product of powers property, and this just says if we have a base number raised to the n power and we multiply that to the same base number to the m power, we can rewrite this then as a raised to the sum of those two powers, n plus m. Then we have the power of a product property, which says that if we have a number that is made up of different factors, so ab, and we raise that to the power of n, we can write this as a product of a to the n and b to the n.
Then we have the quotient of powers property, and it's pretty similar to this one up here, but it's just dealing with division. If we have a raise to n and we divide that by a raise to m, so notice our bases are the same. We can rewrite this as a raised to the difference of those two powers, n minus m. And then we have our power of a quotient property, which is similar to the one above it just dealing with division.
So if we have a quotient, a over b, and we raise that to a power n, we can rewrite this as a to the n over b to the n. And then we have one more, and this is the power of a power property, and this says if we have the number a raised to the power of n and we raise all of that to another exponent m, we can write this as our base number raised to the product of these two powers, n times m. Now, the big idea here is that all of these properties of exponents hold true for negative and fractional exponents too.
So we're going to write that down in our notes as the big idea. These properties of exponents hold true for negative and fractional exponents as well. So let's take a look at some examples, then. So here, I have the square root of 7 times the fourth root of seven cubed. Well, I don't even see any exponents-- well, I guess I do see 1, but what I'd like to do first is rewrite this so that there are no radicals at all.
So remember, radicals can be written as fractional exponents. So I'm going to rewrite the square root of 7 as 7 to the 1/2, and then to rewrite the fourth root of 7 cubed as 7 to the 3/4. And remember that the numerator here always corresponds to that power underneath the radical, and then the four here, the denominator, corresponds to that index of the radical.
So now, what property of exponents am I going to use here? Well, I notice that the bases are the same, and they're being multiplied together, so I can add then these two exponents. So this is the product of powers property, right? So our bases are the same and then we just have two different exponents that we can add together.
So I'm going to rewrite this then as 7 raised to a sum of powers. 1/2 plus 3/4, so this property holds true. We're just dealing with fractions. So we have to add these two fractions. So notice that they do not have common denominators, so I need to rewrite one of the fractions so they do have common denominators. So I'm just going to rewrite 1/2 as 2/4.
Now they can be added together. 2/4 plus 3/4 is 5/4, so I can rewrite this expression here that had radicals as 7 to the 5/4 power. Let's look at another example. Here, I have 1 over 4 squared times 1 over 7 squared. So now, which property am I going to use here? Well, I notice first, I'm going to rewrite these involving negative exponents, because that was something that I remember about 1 over a power.
So I'm going to write 1 over 4 squared as 4 to the negative 2, and I'm going to rewrite one over seven squared as 7 to the negative 2. Well, here, I notice my base numbers are not the same, but I do notice that the exponents are. So I'm going to be using the power of a product property, and really, I'm reading this in the other direction. This right here looks like this, so now I'm just going to rewrite it to look like this.
So I'm going to take my 7 and my four and group those together in multiplication, and then apply my exponent of negative 2, and it's that simple. So now, I'm going to rewrite the standards 28. 4 times 7 is 28. Another example. Here, I have 3 to the negative 2/3 power and 3 to the negative 8/3 power. So hmm. Here, I have I have a fractional exponent that is also negative, but that's OK. I can still apply these properties of exponents.
And I notice that I have the same base number, but they're being divided, so I'm going to be using the quotient of powers property. Same base, but two different exponents, so I'm just going to subtract then those two exponents. So here I have 3 raised to the difference of my exponents, negative 2/3 minus negative 8/3. Well, I'm going to rewrite the subtraction of a negative as the addition of a positive, so 3 raised to the negative 2/3 plus positive 8/3.
So now, I can just add these two fractions together. So negative 2/3 plus 8/3 gives me 6/3. So I have 3 to the 6/3. And 6 over 3 simplifies to 2, so I have simply 3 squared. I can also write this as 9 if you'd like. That's fine too.
Another example here. I have three raised to the 5/7 power, and I'm dividing that by 4 to the 5/7. So let's take a look here. I do not have the same base, but I have the same exponent. So I'm going to be using the power of a quotient property, and once again, it looks more like this side of the equation, and then I'm just going to rewrite it to make it look like this side of the equation.
So I have 3 raised to an exponent power divided by 4 raised to that exponent power. So first, I'm going to take 3 and a 4 and just write that as a fraction, 3 over 4. Group this, and put this to our fractional exponent, 5/7, and it's as simple as that. And let's go through one final example. Here, I have 6 raised to the negative 2, and I'm applying that to another exponent of 3.
So I'm going to be using then the power of a power property. So I'm going to take the product of those two individual exponents and apply that to our base of 6. So I'm going to rewrite this then as 6 raised to the negative 2 times 3 power. The negative 2 times 3 is negative 6, so this can be written as 6 to the negative 6.
So let's go ahead and review our notes. Today, the big idea was that our properties of exponents also hold true if the exponents are fractions or if the exponents are negative numbers. Now, we saw in at least in one example, we had negative fractions. That's OK. These properties of exponents work, and here are those different properties of exponents.
So thanks for watching this video on the properties of fractional and negative exponents. Hope to see you next time.
