Radicals and rational exponents — Harder example

- [Tutor] We're told if 2x - 3y is equal to four, what is the value of four to the X power divided by eight to the Y power? Pause this video and see if you can figure this out. All right, so at first this looks a little bit tricky. You're like, how do I manipulate what I have here on the left to get what I have here on the right? But another way to approach that is to say, look, this thing on the right looks a little bit suspicious, four and eight they aren't... Eight isn't a power of four, but we know that they are both powers of two. And so, maybe we can re-express four as a power of two, and we can re-express eight as a power of two. And maybe if we algebraically manipulate that this might show up, so let's see what happens. So, I'm just going to rewrite everything. So, we have four to the X power over eight to the Y power. Now, as I just mentioned, four is the same thing as two squared? So, we can rewrite this as two squared and then that's to the X power over instead of eight we know that eight is the same thing as two to the third power, and all of that to the Y power. Now, if we know, we know already from our exponent properties, and if this is unfamiliar to you, you can review it on Khan Academy. If you raise something to an exponent and then raise that to another exponent, that's equivalent to multiplying the exponents. So, this is going to be equal to, and I'm gonna get a new color here. This whole numerator is going to be equal to two to the two times X power or two to the two X power and that's going to be divided by, and then this entire denominator right over here, it's going to be two to the third to the Y. So, it's going to be two to the three times y power. Two to the three Y power. Now, we have the same base and we can use other exponent properties. You might recognize that if I have A to the X over A to the Y, this is the same thing as A to the X minus Y. And we explain the intuition of that in other videos on Khan Academy, but we can use that property right over here. We have the same base, and so, this is going to be equal to two that same base to the 2x - 3y power minus we have our 3y over here minus 3y power. And so, this whole thing has been remanipulated or manipulated to be two to the 2x - 3y power and say, where do I go from here? Well, we just have to remember they told us that 2x - 3y is equal to four. So, all of this business is equal to four. So, it's two to the fourth power. Well, we're in the homestretch now deserve a little bit of a drum roll. This is equal to 16 and we are done.