Question Video: Finding Values of Expressions Using Functions and Exponents Mathematics

Given that ๐‘“โ‚(๐‘ฅ) = 8^(๐‘ฅ) and ๐‘“โ‚‚(๐‘ฅ) = (1/8)^(๐‘ฅ), determine the value of (๐‘“โ‚(9/2) โˆ’ ๐‘“โ‚‚(โˆ’4))/(๐‘“โ‚(5/2) โˆ’ ๐‘“โ‚‚(โˆ’2)).

03:44

Video Transcript

Given that ๐‘“ sub one of ๐‘ฅ equals eight to the power of ๐‘ฅ and ๐‘“ sub two of ๐‘ฅ equals one over eight to the power of ๐‘ฅ, 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, weโ€™re gonna substitute in nine over two in for ๐‘ฅ in our function, which is ๐‘“ sub one. Then weโ€™re gonna get minus one over eight to the power of negative four because weโ€™ve substituted that in for our ๐‘ฅ. And then weโ€™ve got eight to the power of five over two because weโ€™ve substituted in five over two for ๐‘ฅ 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 ๐‘ฅ to the power of negative one, this is gonna be equal to one over ๐‘ฅ. So therefore, we can rewrite what weโ€™ve 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 ๐‘ฅ to the power of ๐‘Ž to the power of ๐‘, this is equal to ๐‘ฅ to the power of ๐‘Ž๐‘. So we multiply our exponents. So what weโ€™re gonna get is eight to the power of nine over two minus eight to the power of four. And thatโ€™s 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 itโ€™s at this point we might be wondering what weโ€™re going to do. Well, itโ€™s here that we need to notice something. And that is that we can factor some of the terms that weโ€™ve got within our expression. What we can see is that weโ€™ve got actually a factor thatโ€™s 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, weโ€™ve got eight to the power of nine over two multiplied by one minus eight to the power of negative a half. Well, if weโ€™re gonna multiply eight to the power of nine over two and eight to the power of negative a half, then what we do is weโ€™d 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, weโ€™ve 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, weโ€™re 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 whatโ€™s 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 weโ€™re 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 ๐‘ฅ to the power of ๐‘Ž divided by ๐‘ฅ to the power of ๐‘ equals ๐‘ฅ to the power of ๐‘Ž minus ๐‘. So what weโ€™re 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, weโ€™ve got eight to the power of two or eight squared. So therefore, we can say that the value of our expression is 64.

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