Simplifying hairy exponent expressions

So let's get some practice simplifying hairy expressions that have exponents in them. We have a hairy expression right over here, and I encourage you to pause the video and see if you can rewrite this in a simpler way.

All right, let's work through this together. The first thing that jumps out at me is the numerator. Here, I have the number 125 raised to the 1/8 power times the same number, the same base 125 raised to the 5/8 power. So I can rewrite this numerator. I can rewrite this numerator using what I know of exponent properties as being equal to 125 to the sum of these two exponents: 1/8 power plus 5/8 power. All of that is going to be over the existing denominator we have, which is 5 to the 2.

All of that is going to be over 5 to the 1/2 power. So these are equivalent. Notice all I did is I added the exponents of these two exponents because I had the same base, and we were taking the product of both of these: 125 to the negative 1/8 and 125 to the 5/8. So negative 1/8 plus 5/8, well, that is 2.

So this right over here is 125 to the 2 over 5 to the 1/2. Well, that's going to be the same thing. This is going to be equivalent to 125 over 5, raised to the 1/2 power to the 1/2 power. If I raise 125 to the 1/2 and I'm dividing by 5 to the 1/2, that's the same thing as doing the division first and then raising that to the 1/2 power.

Well, what's 125 divided by 5? Well, that's just 25. And what's 25 to the 1/2? Well, that's the same thing as the principal square root of 25, which is equal to 5. And we're all done. That simplified quite nicely.

Let's do another one of these, and this one is a little more interesting because we are starting to involve a variable. We have the variable W, but it's really going to be somewhat the same process. Here, the thing that jumps out at me is the denominator. I have the same base, 3W^2, raised to one power, one exponent, times the same base, 3W^2, raised to another power.

So this is going to be equal to our numerator. We can distribute it: 12W^7 over -3. Over our denominator, we can write this base 3W^2 times 3W^2, and we can add these two exponents. So we could add -2/3 to -56.

Well, what is that going to be? Let's see if I do negative -2/3 is the same thing as -4/6 minus 5/6, which is equal to -9/6, which is equal to -3/2. So this right over here is the same thing as -3/2. Let me just write that -3 power -2/3 plus -5/6 is -3/2.

Now what's interesting is I have a negative 3/2 up here, and I have a negative 3/2 over here. So we can do the same thing we did in the last problem. This could simplify to 12W^7 over 3W^2 times 3W^2, all of that to the negative 3/2 power. Notice what we did here: I had something to the 3 divided by something else to the 3.

Well, that's the same thing as doing the division first and then raising that quotient to the -3. What's nice about this? This is pretty straightforward to simplify. 12/3 is 4, and W^7 divided by W^2 can be simplified as well. We could divide both by W^2, or you could say this is the same thing as W to the 7 minus 2 power.

So this is going to be W to the 5th power. So it all simplified to 4W to the 5th power to the negative 3/2. Now, if we want to, this is already pretty simple, and at some point, it becomes somewhat someone's opinion on which expression is simpler than another. It might depend on what you're using the expression for, but one could argue that you could keep trying to simplify this.

This is the same thing as 4 to the negative 3/2 times W to the 5 to the negative 3. Once again, this is just straight out of our exponent properties. Now, 4 to the negative 3/2, let's just think about that.

4 to the negative 3/2 is equal to, that's the same thing as 1 over 4 to the 3/2. Let's see: the square root of 4 is 2, and we raise that to the third power. It's going to be 8, so this is equal to 1/8.

So all of this is going to be equal to 1/8, and then W to the 5th and then that to the negative 3/2. We can multiply these exponents. That's going to be W to the 5 times -3, what's going to be the negative 15/2 power.

So I don't know which one you would say is simpler: this one over here or this one over here, but they are equivalent, and they both are a lot simpler than where we started.