Okay, so in this video we’re going to have a look at lots of different problems using indices. Now when it comes to problem solving, there are obviously lots of different types of questions that you can get and there are different methods that you can go through. So what we’re going to have a look at is three different styles of questions throughout this video, starting with the one that you can see on the screen and hopefully, by the end of the video you’re, going to have a little bit of a different toolkit to be able to tackle some of these sorts of questions and maybe even apply some of those methods onto other questions! So we’re going to have a look at three different types and with that being said, let’s get started, [music], okay. So moving on to our first question now, this question here says find the value of, and then we have the fourth root of 27 times 3 times 10 to the power of 8. . Now this is a non-calculator question, so we’re not just going to type this into a calculator we’re actually going to need to work it out without one. So something that we can think about here is actually looking at the base numbers and what i mean by that is? If we look at the start of some of these pieces, we have a 27 and a 3. . Now 27 is actually a power of 3!
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We could write 27 as 3 to the power of 3 as it is 3 cubed. So we’ve got three to the power of three multiplied by three to the power of one.
So actually that could simplify down to one power of three and if we add those powers together, that would become three to the power of four. Remembering you add the powers when you are multiplying as long as we have the same base numbers that is which we do so now we have 3 to the power of 4. We’ve then got that that is being multiplied by 10 to the power of 8.
, so we’ve simplified the first bit and now we know it’s 3 to the power 4 multiplied by 10 to the power of 8. ! Now what you could do is you could actually go and work that out. Okay, you could work out 3 to the power 4. You could work out 10 to the power of 8. You could add all those zeros on, but ultimately we are also going to have to find the fourth root of that which isn’t going to be very nice! So now we need to think about that fourth root and how we could actually apply something a little different for writing the fourth root. Now we should already know that when we write a certain power like something like x to the power of a half x to the power of a half is the same as the square root of x. I’m going to put that two in place, because obviously, here we’re looking at a fourth root? So if we instead we’re going to have a fourth root there, we could think about what power would that would be well, it would be x to the power of a quarter and that would actually find the fourth root, but how does that help us? so if, instead, we write this as alt to the power of a quarter and again, we’ll then think about how we could then apply this to the powers? Well, if we write it all to the power of a quarter when we were previously dealing with indices, where there’s brackets involved, we know that something like x to the power of a to the power of b outside the bracket! You multiply those powers and that would become x to the power of a b!
Well, if we apply that to these fractions here, we just have to multiply the powers and we’ve got a four and we’ve got an eight that both need multiplying by a quarter and again, there’s something else to be thinking about here, because when you multiply by a quarter, what does that actually do? well, it finds a quarter of that number. For example, if we took something like 8 and we multiplied it by a quarter and picked 8, because that’s one of the numbers we’re going to do, we would get the answer 8 over 4, which is 2. Okay and a quarter of 8 is 2, so it just finds a quarter for us. So if we apply that onto these, we would have 3 to the power of, and a quarter of four would be one and then we would have the ten and a quarter of the power of eight we’ve already worked out is two? So we have three to the power of one times ten to the power of two, and that is now something that we can actually go ahead and work out, because that’s three times 10 squared that’s 3 times 100. So the final answer there would just be 300 and we don’t have to do any more work now and we’ve dealt with all those powers. So there’s a lot going on there. So let’s just have a very quick recap! So to start with, we thought about collecting together those numbers with the same base number!
So we turned 27 into three cubed. We were then able to add that to the three to the power one or add together those powers so that we had three to the power of four. We were then able to think about the fourth root in a different way, so putting it to the power of a quarter instead and how that how that affected the powers when we multiplied out the brackets, so the power of 4 was quartered to become 1 and that power of 8 was quartered to become 2.
. We were then just able to work that out so 3 times, 10 squared okay. So a pretty tricky question.
You can see, obviously why this is problem solving, because these are a little bit trickier than normal questions, but i’ve got a couple of these for you to have a go at? So let’s have a look at those now. Okay, so there’s two questions here for you to have a go out so pause. The video have a go at these two and we’ll give the answers in a sec right so for the first one. Then now, if we simplify or join together, these numbers to start with we’ve got 2 to the power of 1 multiplied by and we can write 8 as 2 cubed. Now that’s going to become 2 to the power of 4 when we add together those powers and we’re multiplying that by 10 to the power of 12. . Now we can think about again the fourth root which could be written as a power of a quarter, and if we write that as a power of a quarter, it’s going to quarter those powers, a quarter of four would become two to the power of one and a quarter of twelve would become ten to the power of three and there we go now. We can work that out. That is two multiplied by a thousand and two multiplied by a thousand gives us the answer.
Two thousand, and there is our first answer.
We get the answer two thousand on to the next one. Let’s have a look at this approaching it in the same way. If we join these together, we’ve got four to the power of one multiplied by and that sixteen could be written as four squared. Now again, you could potentially turn this into powers of two, and that would be fine as well, but we may as well just stick with powers of four here, as we’ve got the same base number with those anyway, so that becomes four to the power of three and we’re going to multiply that by the 10 to the power of 15? .
Now this time we have a cube root. So, instead of writing a power of a quarter, that’s going to be a power of a third and that’s going to find the third of each of those powers.
So the power of 3 becomes 4 to the power of 1, and the power of 15 is going to become 10 to the power of five. So we’ve got quite a large power of ten there, and that is going to be four multiplied by and how many zeros would be there well ten times? Ten is a hundred a thousand ten thousand one hundred thousand. So we’ve got four times? One hundred thousand, which is going to be four hundred thousand and there we go, and there is our final answer- and that is our first type of indicy problem?
We’re gonna have a look at in this video right. Okay, so let’s have a look at a different type of indices problem. Okay, so this one is involving a little bit of equations as well!
So it says, solve 3 to the power of 2x is equal to 1 over 81. !
Now, for the purpose of this one, we’re going to have to think about how what is another way of writing 1 over 81 and we’ve got 3 to the power of 2x there. So if we can write 1 over 81 as a power of 3, then potentially we might be able to have a look at them. Now.
If we can say that this piece on the right is 3 to the power of something, then we can automatically have a look at those two powers, because we’ve got 3 to the power of 2x is equal to 3, to the power of something, and we know those powers have to be the same!
So we’ll deal with that when we get there, but this is the kind of problem we’re gonna have a look at here! Can we write this piece on the right as a power of three now the answer has to be yes, because 81 is a power of three and if you might not know that, but if you work that out three times three is 9 times: 3 again is 27 times. 3 again is 81.
, so 81 is 3 to the power of 4! , so we could write 81 as 3 to the power of 4? Okay. So let’s just write that down, but of course this is an 81? This is 1 over 81. .
So in order to get that to do the reciprocal to make it 1 over 81, i would have to write it as 3 to the power of negative 4 and 3 to the power of negative 4 is 1 over 81! . So there we go. That’s how we’re going to get it to a point where we can think about how we would write 1 over 81 is 3 to the power of 4. .
If we now take that we can set them equal to each other, so we have 3 to the power of 2x, which is equal to 3, to the power of negative 4, and we can now almost forget about the base numbers there, because we know that that power of 2x has to be equal to the negative 4. So we can actually just write that as an equation, and we can just say 2x is equal to negative 4 and then solve it from there well 2 times a number is equal to negative 4 and if you divide by 2, you get x is equal to negative 2! And there we go we’ve solved it. We’ve got x is equal to negative 2. That is the power that would have to go in there to ensure that we get 1 over 81, because 3 to the power of negative 4 is equal to 81 over 81. . So there we go.
That is another type of problem, so let’s have a little look and some questions on this for you to have a go at okay, so there’s two questions very similar to the ones we just looked at so have a go at these two pause. The video there we’ll give the answers in a sec right so for the first one, then so 16 so 16 is going to be 2 to the power of 4? ! So if 16 is 2 to the power of 4, then 1 over 16 would be equal to 2 to the power of minus 4. . So now we’ve got them both as 2 to the power of something we can write. Our equation, 2 to the power of 2x, has to be equal to two to the power of negative four! Just like the previous question, we can set those powers equal to each other, so two x is equal to negative four and we can divide by two x is equal to negative two and there we go and there’s our first one now the second one’s a little bit trickier than that one! So, let’s have a look, so we’ve got five to the power of six x and we’ve got one over 125. . Now 125 is 5 to the power of 3. .
So we can write that as 5 to the power of 3, and that would mean that 1 over 125 is equal to 5 to the power of negative 3. . So there we go, we’ve got it as 5 to the power of 3 or 5 to the power of negative 3. Now so we can set them equal to each other! 5 to the power of 6x is equal to 5, to the power of negative 3, and if we set those powers equal to each other, we have 6x that has to be equal to negative three, and you can divide by six. So x is equal to negative three divided by six, which is equal to minus a half? When we simplify that fraction and there we go and there’s our final answer x is equal to minus a half! So there we go another couple of problems, and hopefully that was helpful, something a little bit different there and problem solving with indices?
When we have to actually create the negative power by looking at the reciprocal right, let’s have a look at our final problems. Okay, so these questions are the hardest out of the bunch. It says, given that 3 to the power of negative n is 0?
2, find the value of 3 to the power of 4 to the power of n! Now, for starters, here we need to figure out what n is so that we can actually think about how we would approach this now. We might not actually need to find out what n is, but that’s certainly where i thought our first train of thought should go with this, but we know that 3 to the power of negative n, just from those previous questions means that we could write 0!
2 in a different way. So, instead of writing, 3 to the power of negative n is equal to 0. 2. Could we get rid of the negative power and instead write the reciprocal? and yes, we can.
We can write 3 to the power of positive n would therefore be equal to 1, divided by 0. 2, the reciprocal of 0. 2. Now that would be a strange way of writing that reciprocal, because 0. 2 is a decimal! We shouldn’t have decimals within fractions, so we should actually work this out and we can actually divide 0. 2 into one there’s a couple of different ways that you could do that either you’re going to know how many times naught point two fits into one or you could simplify the fraction by multiplying the top and bottom by five to make the bud to make the denominator there become one, and that would give us three to the power of n is equal to and it’d be five over one? So the answer is five: 0. 2 does fit into one five times, so we’ve got to a specific point here: we’ve got three to the power of n equals five. Now at this point, you might be looking at it still and thinking! Well, we can’t figure that out?
I don’t know what power of three would give me. The answer. Five and you’d be right to think that, because actually, what we need to do is look at this second piece of information! Now it’s given to you in a bit of a strange way because it says 3 to the power of 4 to the power of n, but that is actually a bit of a bit of a trick there to try and confuse you, because actually, if we have something to a power, let’s say x, to the power of a all to the power of b, those two powers just get multiplied and we get x to the power of a b! But does it matter if it was written as x to the power of b all to the power of a because ultimately, we would still get x to the power of a b. So for this one here we don’t actually need to write the powers that way round? We could actually write them. The other way around. We could say that 3 to the power of n, all to the power of 4 is exactly the same thing? So let’s get rid of this because we don’t need that. But ultimately, we now have an expression where it says 3 to the power of n to the power of 4 and that’s what we’re finding the value of now. We’ve actually just worked out. What 3 to the power of n is actually equal to it’s equal to 5! . So, instead of writing 3 to the power of n in the bracket, we could just put 5 in there, because we know that 3 to the power of n is equal to five, which we’ve just got from there, and we want that to the power of four. So we just need to work that out. We just need to do five times five times five times: five, which again without a calculator, isn’t very nice five times five is 25 times! 5 again is 125 times?
5 again is 625, so the answer for that is 625 and there we go and there we are- we’ve solved it, so that was probably the hardest out of the bunch, because we’ve actually done the reciprocal of the other side to get rid of that negative power!
In the first step. Just here, then, we have actually had to work that out so 1 divided by 0. 2, and that gave us a value of 3 to the power of n. We then did a little switch of the powers just here, so that we were able to actually substitute the number five in and work out that value! So there’s quite a lot going on, and i have got a couple of these for you to have a practice on and have a little go out? You’re definitely going to need to practice some of these, but they will be our last few questions for this video.
So let’s have a look at those now.
Okay, so there’s two questions here so pause. The video see if you can get an answer for both of these and we’ll go over the answer in just a sec right so for the first one then so 5 to the power of negative n. Now we could write that as 5 to the power of n is going to be equal to 1!
Over 0! 5 1, divided by 0! 5, is 2, so 5 to the power of n is equal to 2, and now we can do the little switch of the powers so 5 to the power of n all to the power of 3, and we know that 5 to the power of n is equal to 2, so that is going to be 2 in the bracket to the power of 3 and 2 to the power of 3 is equal to 8. ! So there we go there’s our answer: eight, not too bad.
Hopefully you were okay with that on to the next one, we’ve got seven to the power of negative n is 0. 1, so let’s get rid of that negative power. We’ll do the reciprocal of 0.
1, so 1 over 0! 1 1, divided by 0? 1, is 10.
So 7 to the power of n is equal to 10, and now we can do our little power swap. So, instead of that, we’ll write 7 to the power of n all to the power of 4, and now we can substitute our value of 7 to the power of n into our bracket and we get instead. 10 in the bracket to the power of 4 and 10 to the power of 4 is 10 times 10 times 10 times 10, which is 10 000. And there we go, and there is our value! We’ve got our answer and there we go. We’ve solved all our problems, so there were three different types of question there, hopefully they’re all useful? You may need to go back and watch them and have another go at some of these, but hopefully that was useful and helpful if it was please like please comment, please subscribe and i’ll see you for the next video [music] uh [music]. .