### Video Transcript

Find the fourth term in the expansion of five over root ๐ฅ plus root ๐ฅ over five to the power of nine.

So, what we can use here is the binomial expansion and we have a general form for that. If we have ๐ plus ๐ to the power of ๐, then we can say that if weโre gonna expand this, itโs ๐ choose zero ๐ to the power of ๐ ๐ to the power of zero plus ๐ choose one ๐ to the power of ๐ minus one ๐ to the power of one et cetera up to ๐ choose ๐ ๐ to the power of zero ๐ to the power of ๐. So, whatโs happening is the first term, you can see that, actually, the exponent is decreasing by one each time. And for the second term, the exponent is increasing by one each time.

So, in our question, what weโre looking for is ๐ sub four which is the fourth term. Well, we could do this a couple of ways. If we take a look at our general form for the binomial expansion, we could see that for the first term, itโs ๐ choose zero ๐ to the power of ๐ ๐ to the power of zero. Then, the second term is ๐ choose one ๐ to the power of ๐ minus one ๐ to the power of one. Well, if we take a look at our ๐ choose zero and ๐ choose one, then our ๐ is going to be nine because thatโs the power that the parentheses are raised to. So, weโve got nine. Then, weโve got choose three. And thatโs because the bottom number is always one less than the term number. And then, weโre gonna have ๐ to the power of six. And thatโs because if we take a look at the expansion, the first ๐ would be ๐ to the power of nine. Then, weโd have ๐ to the power of eight, and then two further down the line would be ๐ to the power of six. And then, weโre gonna have ๐ to the power of three. And thatโs because the exponent or power of ๐ is always one less than the term value. And also, as a quick check, if you add the exponents, they should add up to ๐. So, six add three is nine, yes, and ๐ is nine, great.

So now, letโs apply this to our expansion to find out what the fourth term is going to be. So, if we identify our ๐ and our ๐, our ๐ is five over root ๐ฅ and our ๐ is root ๐ฅ over five. So, what weโre gonna have is ๐ sub four or the fourth term is equal to nine choose three multiplied by five over root ๐ฅ all to the power of six multiplied by root ๐ฅ over five all to the power of three.

So, first of all, what we want to work out is what is the value of nine choose three. Well, nine choose three is equal to 84. And to find out what we can do is in our calculator, we press nine. And then, thereโs an nCr button that youโll find. Often, itโs you have to press shift and then a button to find nCr and then three. And thatโll give us 84. And then, what Iโve done for the other two terms is that Iโve changed them to exponent form. So, weโve got five ๐ฅ to the power of negative a half all to the power of six multiplied by ๐ฅ to the power of a half over five all to the power of three. And we got that using a couple of exponent rules. And that is that if we have ๐ฅ to the power of a half, itโs equal to root ๐ฅ. And if you have ๐ฅ to the power of negative one, itโs equal to one over ๐ฅ.

So then, what we have is the fourth term is equal to 84 multiplied by 15625๐ฅ to the power of negative three. And thatโs because weโve raised five to the power of six, which is 15625. And then, weโve got ๐ฅ to the power of negative a half. Well, if we raise this to the power of six, then what we do is multiplying the exponents and we get negative three. And then, this is multiplied by ๐ฅ to the power of three over two over 125. And as weโve already said, itโs because we used another one of our exponent rules. And that is, if youโve got ๐ฅ to the power of ๐ to the power of ๐, then all we do is multiplying the exponents. So, we get ๐ฅ to the power of ๐๐.

Okay, great. So now, what we need to do is another step of simplifying. So first of all, weโre gonna get 10500. And thatโs because itโs 84 multiplied by 15625 divided by 125. And then, weโll have ๐ฅ to the power of negative three over two or negative three-halves. And thatโs because we use one more exponent rule. And that is if we have ๐ฅ to the power of ๐ multiplied by ๐ฅ to the power of ๐ is equal to ๐ฅ to the power of ๐ plus ๐. And if we have negative three and add on three over two or one and a half, weโre gonna get negative three over two.

So therefore, we can say that the fourth term of our expansion is going to be 10500๐ฅ to the power of negative three over two.