### Video Transcript

Part a) Simplify ๐ก squared multiplied by ๐ก cubed. Part b) Solve five minus ๐ equals 11. And Part c) Solve a half multiplied by ๐ minus three equals six.

So in part a, what we actually have is ๐ก squared in brackets next to ๐ก cubed in
brackets. And what this actually means is ๐ก squared multiplied by ๐ก cubed. And therefore, to actually solve this problem, what weโre gonna have to use is one of
our index laws. And the index law weโre gonna use is one that tells us that if we have ๐ฅ to the
power of ๐ multiplied by ๐ฅ to the power of ๐, then itโs equal to ๐ฅ to the power
of ๐ plus ๐.

So what we do is we actually add the powers. The key to actually allow this to work โ so this index rule to work โ is to have the
same base numbers. So we have so. We have ๐ฅ and ๐ฅ in our index law. And if we look to our question, our bases are actually ๐ก and ๐ก. So we have the same base. So we can apply this law.

And when we do that, we get ๐ก to the power of two plus three. And thatโs because two was like our ๐ and three was like our ๐ and we add them
together. So therefore, weโre going to get an answer of ๐ก to the power of five. So we can say that ๐ก squared multiplied by ๐ก cubed fully simplified is equal to ๐ก
to the power of five.

Okay, great, but letโs have a look at why is this. So why does this work? Why does this index law solve this kind of problem? Well, weโre gonna start off by having a look at ๐ก squared. What ๐ก squared actually means is ๐ก multiplied by ๐ก. And in the question, it said letโs multiply this by ๐ก cubed. So Iโve done this and actually what Iโve got now is ๐ก multiplied by ๐ก โ so our ๐ก
squared โ then multiply it by ๐ก multiplied by ๐ก multiplied by ๐ก, our ๐ก
cubed. So as we can see, we actually have now five ๐กs. And theyโre actually all multiplied together, which gives us ๐ก to the power of
five.

So now, we can see why this index law works. Okay, great, letโs move on to part b. So in part b, weโre trying to solve an equation. So we have five minus ๐ is equal to 11.

Now, the first stage in order to do that is to actually add on ๐. And the reason weโre doing that is because first of all we want to make the ๐
positive because itโs easier to deal with. And it also means that actually thereโs less chance of making an error with negative
numbers. Then, this gives us five is equal to 11 plus ๐. And thatโs because if we have negative ๐ plus ๐, it gives us just a zero. So thatโs why the left-hand side is just five and then we just add ๐ to the
right-hand side.

So now, to actually find the value of ๐, what we need to do is actually leave the ๐
on its own. And to do that, what weโre gonna do is actually subtract 11 from each side of the
equation. Therefore, if we do that, we get negative six is equal to ๐. And thatโs because five minus 11 gives us negative six. And if we have 11 plus ๐ and then we subtract 11 from that, we just get left with
our ๐.

Itโs also worth mentioning at this point when we take 11 away from five, I said we
got to negative six. And a quick way of doing that if youโre not quite sure with the negative numbers is
if you think, well, we take five away to get to zero, then thereโs still six left
from 11. So you take that six away. We get to negative six. Okay, great, so we can now say that the solution to five minus ๐ equals 11 is ๐ is
equal to negative six.

What we can now do though is actually carry out a check to make sure that weโve
actually got the correct answer. And to do that, we actually substitute ๐ is equal to negative six back into the
original equation. And when we do that, we get five minus negative six because negative six without ๐
is equal to 11.

And then, we need to remember that if we actually subtract a negative, thatโs gonna
give us a positive. So five minus negative six is gonna be the same as five add six. So then, that leaves us with five plus six is equal to 11, which is great cause itโs
the same as we had in the original equation. So therefore, we can say you fully checked, we know that ๐ is equal to negative
six.

Okay, now, letโs move on to part c. So now, in part c, what we want to do is solve a half multiplied by ๐ minus three
equals six. And the first step is going to be multiplying both sides of the equation by two. And thatโs because weโve got a half ๐ minus three. So therefore, if we have a half multiplied by two, weโll just get ๐ minus three. So itโs easier to deal with.

But of course, whatever we do to this side of the equation, we must do to the other
side of the equation. So now, when weโve actually done that and multiplied both sides of the equation by
two, we get ๐ minus three is equal to 12. So now, itโs actually the value of ๐ that weโre looking to find.

So now, as we want to do that, what weโre gonna do is actually add three to each side
of the equation. The reason we do that is because if we add three to negative three we get zero, so
then it will leave ๐ on its own. And as we said before, whatever you do to one side, we gotta do to the other side of
the equation. So when we do that, we get ๐ is equal to 15. And thatโs because 12 add three is equal to 15.

So now, we found the solution to a half ๐ minus three equals six and that is ๐ is
equal to 15. Well, what we can do now โ like weโve done before โ is actually check to make sure
this is correct. And the way we do that is by substituting in ๐ is equal to 15 back into the original
equation. And when we do that, we get a half multiplied by 15 minus three is equal to six. So then, what we get is a half multiplied by 12 is equal to six. Well, this means half of 12 is six. So yes, this is correct. And itโs the same value that we got with the original equation.

So we can say that weโve now solved part a, b, and c. And the answers are ๐ก to the power of five, ๐ equals negative six, and ๐ equals
15, respectively.