### Video Transcript

Given that ๐ sub one of ๐ฅ equals eight to the power of ๐ฅ and ๐ sub two of ๐ฅ equals one over eight to the power of ๐ฅ, determine the value of ๐ sub one of nine over two minus ๐ sub two of negative four over ๐ sub one of five over two minus ๐ sub two of negative two.

So our first term is gonna be eight to the power of nine over two. We get that because we have ๐ sub one of nine over two. So therefore, weโre gonna substitute in nine over two in for ๐ฅ in our function, which is ๐ sub one. Then weโre gonna get minus one over eight to the power of negative four because weโve substituted that in for our ๐ฅ. And then weโve got eight to the power of five over two because weโve substituted in five over two for ๐ฅ and then minus one over eight to the power of negative two. Okay, great! So what do we do now?

Well, now what we need to do is start to use some of our index laws. Well, the first one is that if we have ๐ฅ to the power of negative one, this is gonna be equal to one over ๐ฅ. So therefore, we can rewrite what weโve got as eight to the power of nine over two minus eight to the power of negative one all to the power of negative four divided by eight to the power of five over two minus eight to the power of negative one to the power of negative two.

So then what we can do is apply another one of our index or exponent rules. And that is if we have ๐ฅ to the power of ๐ to the power of ๐, this is equal to ๐ฅ to the power of ๐๐. So we multiply our exponents. So what weโre gonna get is eight to the power of nine over two minus eight to the power of four. And thatโs cause we have negative one multiplied by negative four, which gives us four, then all divided by eight to the power of five over two minus eight squared.

So itโs at this point we might be wondering what weโre going to do. Well, itโs here that we need to notice something. And that is that we can factor some of the terms that weโve got within our expression. What we can see is that weโve got actually a factor thatโs the same in the numerator and denominator. And that is one minus eight to the power of negative a half. Well, how do we get that?

Well, if you take a look at the numerator, weโve got eight to the power of nine over two multiplied by one minus eight to the power of negative a half. Well, if weโre gonna multiply eight to the power of nine over two and eight to the power of negative a half, then what we do is weโd add the exponents. Well, nine over two plus negative one over two gives us eight over two, which is four, which is what we had on the numerator previously.

So then if we look at the denominator, weโve got eight to the power of five over two multiplied by one minus eight to the power of negative a half. So again, if we had our exponents, weโre gonna have five over two plus negative a half, which gives us four over two or two, again what we had originally. Okay, great! So now whatโs our next step?

Well, the one minus eight to the power of negative a half is gonna cancel from the numerator and denominator cause we can divide through by this. So what weโre gonna have is eight to the power of nine over two divided by eight to the power of five over two. So then what we can do is use another one of our index laws. And that is that ๐ฅ to the power of ๐ divided by ๐ฅ to the power of ๐ equals ๐ฅ to the power of ๐ minus ๐. So what weโre gonna get is eight to the power of nine over two minus five over two. Well, nine over two is the same as four and a half. And five over two is the same as two and a half. So four and a half minus two and a half gives us just two.

So therefore, weโve got eight to the power of two or eight squared. So therefore, we can say that the value of our expression is 64.