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Simplifying hairy expression with fractional exponents

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- [Voiceover] So let's get some practice simplifying hairy expressions that have exponents in them. And so 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 and 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. To the negative 1/8 power plus 5/8 power. All of that is going to be over the existing denominator we have, which is five to the 1/2. All of that is going to be over five to the 1/2 power. So, these are equivalent. Notice, all I did is I added the exponents, 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. And so negative 1/8 plus 5/8, well that is 1/2. So, this is, this right over here is, 1/2. So, this is 125 to the 1/2, over five to the 1/2. Well, that's going to be the same thing. This is going to be equivalent to 125 over five, over five, to the 1/2 power. To the 1/2 power. If I raise 125 to the 1/2 and I'm dividing by five 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 five? Well, that's just 25. Now what's 25 to the 1/2? Well, that's the same thing as the principal square root of 25, which is equal to five. And we're all done, that simplified quite nicely. Let's do another one of these. And this one is a little more interesting, 'cause we are starting to involve a variable, we have the variable w, but it's really going to be somewhat the same process. And here the thing that jumps out at me is the denominator. I have the same base, 3w-squared. 3w-squared raised to one power, one exponent, times the same base, 3w-squared, raised to another power. So, this is going to be equal to, this is going to be equal to our numerator we can just rewrite it 12w to the seventh power over negative 3/2 over our denominator we can write as this base, 3w-squared, and we can add these two exponents. So, we could add negative 2/3 to negative 5/6. Negative 2/3 to negative 5/6. Well, what is that going to be? So, let's see, if I do negative, negative 2/3 is the same thing as negative 4/6, minus 5/6, which is equal to negative 9/6, which is equal to negative 3/2. So, this right over here is the same thing as negative 3/2. Let me just write that. Negative 3/2 power. Negative 2/3 plus negative 5/6 is negative 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, 12w to the seventh power, over 3w-squared. 3w-squared. All of that, all of that to the, all of that to the negative 3/2 power. Notice what we did here. I had something to the negative 3/2 divided by something else to the negative 3/2. Well, that's the same thing as doing the division first and then raising that quotient to the negative 3/2. And what's nice about this, this is pretty straightforward to simplify. 12 divided by three is four, and w to the seventh divided by w-squared, well, we could divide both by w-squared or you could say this is the same thing as w to the seven-minus-two power. So this is going to be w to the fifth power. W to the fifth power. And so it all simplified to 4w to the fifth power to the negative 3/2. 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 is expression is simpler than another, and 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, this is the same thing as four to the negative 3/2. To the negative 3/2. Times w to the fifth to the negative 3/2. And once again this is just straight out of our exponent properties. Now, four to the negative 3/2, let's just think about that. Four to the negative 3/2. We're using a different color just as a bit of an aside. So, four to the negative 3/2 is equal to, that's the same thing as one over four to the 3/2, and, let's see. The square root of four is two and then we raise that to the third power, it's gonna be eight. So, this is equal to 1/8. So, that's equal to 1/8, and so all of this is going to be equal to 1/8. And then w to the fifth, and then that to the negative 3/2, we can multiply these exponents. That's going to be w to the five times negative 3/2. Well, it's gonna 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.