Negative indices

Key Stage 3

Maths

Indices and standard form

﻿Hello, and welcome to this lesson on negative indices with me, Miss Oreyomi. For today's lesson, you will be needing a paper and a pen or something you could write on and with. If you need to put yourself in a space with less distraction so that you can concentrate for the duration of this lesson, then please do so. Also, if it helps, if it would help you by putting your phone on silent, also please do so. If you need to pause the video now to get these things done, then again, do so and press record when you are ready to begin the lesson. Okay, for your try this task, you are to fill the boxes, these empty boxes here, with the green cards to make as many true statements as possible. So for example, if I take this and I take that, well, my answer because I am dividing my indices, I'm subtracting, so therefore it's going to be five. So I have used three, I have used negative two and I have used five. So your job now is to fill in the boxes with the green cards to make as many statements that are true, many statements that are correct. So pause the video now and attempt this, and then once you're done, press resume to carry on with the lesson. Okay, hopefully you had a go at that. I'm just going to go through some more examples. So say for example, I say zero and then I just put a negative one here, well, I am going to have one here. Let's do another one. If I put a four there and I put a six there, well, four subtract six would be a negative two. So I have used my four, my six and my negative two over there again. Let's do one more, if I have a negative six here and then a negative three here, well, it would be negative six take away negative three. So therefore this is going to be a negative three. So I hope you are able to come up with as many, I hope you are able to fill in the boxes with as many green cards to make true statements. From our try this task, we can see our law of indices at work. We can see we use the law of indices that if we're dividing powers with the same base, if we're dividing indices with the same base, then we're subtracting the powers away from each other. So let's try that now. I've got two to the power of three divided by two to the power four. Well I simply have to take my powers away, isn't it? So I've got three take away four. So it's going to be two to the negative one. Is there another way, if I want to take from the index form to a fraction, how can I write that? Well, let's see if this can help us. We know that two to the power of three is eight and two to the power of four is 16. So if I write that as a fraction, eight divided by 16, what am I going to get? I'm going to get one over two, aren't I? So essentially, if I've got to raised to the power of negative one, that would give me, that will be the same as me writing one over two. Let's do another example. I've got four squared, if I'm taking over the powers, my simplified answer is going to be four to the negative two. How can I write this? Let's simplify this out, let's work it out. Four squared is 16, just going to write over here, and four to the power of four is 256. If I write these two as a fraction, 16 divided by 256, what's my answer going to be? Well, that's going to be one, wouldn't it? And that's going to be 16. So four squared divided by four to the power of four is one over 16. Do you see what's happening here? When my base is raised to a negative power, I invert it and turn it into a positive fraction. So this negative becomes, I flip it and it becomes a one over four squared. So the negative makes it a fractional, a fraction. So if I want to do five cubed divided by five to the power of five, what's my answer going to be? Well, three takeaway five is five to the negative two, isn't it? And then if I want to write this as a fraction, what am I going to do? I've got a negative two. So that negative becomes a one over something, one over five squared. Can I simplify this further? Yes, I can. So it's going to be a one over 25. Let's see if it makes sense with our multiplication and our division. If I have five squared, that is 25, if I have five to negative two, that is one over five squared, which is 25. If I have two cubed, that is what? Eight. If I have two to negative three, to the power of negative three, that is going to be one over two cubed, which is the same as one over eight. Do you see the pattern then? When I'm dividing, when I have a negative power, it becomes a fractional answer. So I put a one and then I raise it to my positive power. Let's do one more. If I have five to the five cubed, that is going to give me 125, isn't it? What if I have five to the negative three, five raised to the power of negative three? Well, that's going to be one over five cubed, and that is one over 125. How about you have a turn then? Evaluate the three numbers on your screen. So pause the video now and attempt these three questions, and then once you're done press resume and we'll go over it together. Okay, I hope you had a chance of going over that. If I've got eight squared, well, we've established that cause it's eight, to the negative two rather, if I've got eight raised to the power of negative two, I am going to take that negative and make it into a fraction, so it's going to be one over my positive eight squared. And then if I simplify that, this is going to be one over 64. So essentially, if I've got a to the power of negative two, or a to the power of negative n, a being any base, any number, that is the same as saying it's one over a raised to the power of n, n being my power. So that's power, and that's my base. Okay, for the second one then, five raised to the power of negative three. Well, that's going to be one over five cubed, which is the same as one over 125. And then the last example then, four raised to the power of negative two is going to be one over four, one over four squared, which is one over 16. What if we're told then to write this fraction in index form? So I've got to write it in index form, remembering that in index form we'll have our base and our power. So if I want to make this into a base and a power, what's going to be my base? My base is going to be three. And what's going to be my power, what's my power going to be raised to? What's my base going to be raised to, rather. Negative two. Okay, because if I rewrite this back to the form I was given, it's going to be one over three squared, which is going to be one over nine. So when you're told to write in index form, when you're given a fraction, you convert it to a base raised to a power. Let's do this one. This is going to be what? What's my base in this one? Four. And because it's a fraction, it's one over something, it's going to be, my power is going to be a negative number four, like so. What of this one? Write this as a fraction. So I've got 36 raised to the power of negative a half. So again, I start with my fraction, which is one over 36. Remember that because it's a negative I have a one over, and then I raise it to my positive power, a half. If we want to simplify this, 36 raised to a half, we know that this is the same as finding the square root of 36. So my answer as a fraction would be? One over six. Let's try this one. Oh, that should be a negative number there. So I've got 27, because it's a negative number, I'm going to have my fraction, one over 27. And then I'm going to raise my base, 27, to two thirds. Am I done? Not quiet. So, if I simplify it further, I've got one over, this three here tells me I'm taking the cube root of 27 and I'm raising my answer, I'm squaring my answer. Cube root of 27 is? Three, three squared is? Nine. So my final answer is going to be one over nine. Why don't you have a go at these four questions now, try to use the skills we've learned in the previous slide to answer these four questions. So pause the video now and attempt these four questions and press resume when you're ready to go over your answers. Okay, hopefully you had a go at those and hopefully we can now go for answers and see if we got the same thing. So I've got one over nine squared in index form, meaning that I must have a base raised to a power. This one here tells me that my power must be a negative number. So therefore, my base is? My base is nine. And my power is therefore? Negative two. Good job. Okay, next, my base is? Seven, this one tells me my power must be a negative number. So my power is therefore? Negative three. Okay, let's try this one. I want you to write this as a fraction, okay? And again, I've got this negative power, So, to turn it into a fraction, I am going to have one over my base, 64. Because I have written that one over there, I am now raising my power to a positive power, a half. I'm not quite done with my simplification, am I? Not quite. So I am going to do one over the square root of 64, which is one over eight. Obviously you can jump from this step straight there, okay? This one then, I've got one over 84, three quarters. Which is the same as finding the fourth root of 81, 81 not 84. 81, and then cube in my answer. The fourth root of 81 is three, and three cubed is 27, so it's one over 27 over there. It is now time for your independent task. So I would like for you to pause the video now, attempt every single question on the worksheet and then come back and we'll go for the answers together. So pause your video now, and attempt the questions on your sheet. Okay, how did we get on with those? So we want to evaluate the following. So I've got one over three cubed, which is the same as one over 27. Here I've got one over 10 raised to the power of one. Any number raised to the power of one is the same, isn't it? So we've got one over 10. Over here, I've got one over two to the power of five, which is the same as one over 32. Over here, I've got one over six cubed, which is 216. So that is one over 216. Over here, I've got one over two to the power of six. And I believe that is 64, so that's one over 64. Check in your work, making sure you're correcting it if you got it wrong, and ticking your work if you got it right. So over here I've got one over 10 to the four, which is 10,000. Write each of the following in index form. Okay, we know, we know that index form we need our base and our power. Since we have a one over six five, we know that our power must be a negative number. So I'm starting with my base of six, my power is therefore? Negative five, good job. Here I've got four and negative five, four to the power of negative five. Over here, I've got three raised the power of negative four. And over here, I've got five raised of the power of negative three. How did you get on? Did you get all those right? Next one then. I want to write each of the following as a fraction. So, if I've got a to the negative, a raised to the power of minus two, this is going to be the same as saying, a, one over a squared. So this is going to be the same as writing one over a squared. Second one is going to be the same as writing one over m raised to the power one, you don't have to put that one there. This is going to be the same as a writing one over t to the power four. Next, this is going to be the same as writing one over four raised or the power five over two. This is going to be the same as writing one over 100,000 raised of the power of two fifth. And this one is going to be the same as writing one over eight raised to the power of two third. For D, E and F, should we have a go at answering and simplifying those? I think we should. And if you didn't do those, then you can pause the screen now and have a go at those. I'm going to wait for three seconds, if you want to have a go, or you can wait for us to do it together, you can either have a go yourself in three, two and one. So pause the screen now. Okay, so if I want to simplify this, I would have one over the root of four raised to the power of five. So it's going to be one over two, which is the root of four, raised to the power of five is 32. Over here, I would have one over, I believe the fifth root of 100,000 is 10. So 10 squared, so I'm going to write five, 100,000 there and then raised to the power of two. So my answer is going to be 100. Over here we would have one over cube root of eight, and then square that. Cube root of eight is two, two squared is four, and that's our final answers. For your explore task then, you have negative four raised to the power of negative three raised to the power of negative two, raised to the power of negative one. Your task is therefore, can you rearrange these numbers to make the highest possible value? So you have one base and you're raising the rest to a certain power. And then can you rearrange these numbers then to make the lowest possible number. So pause your screen now and have an exploration playing around with the numbers. How can you order this so that you can have the highest possible value, and so that you can have the lowest possible value? If you're feeling confident, pause the video now and have a go. If you want some support, carry on watching the video and I'll provide you with a little bit of support that could help you get started with this task. So pause the video now if you feel like you know what to do. Okay, for your support. So I've got, if I've got negative four raised to the power of three, what does that mean? If I have negative four raised to the power of negative three, what does that mean? If I have positive four raised to the power of three, and then I've got positive four raised to the power negative three. Use this hint to help you work out what your base value should be and what the rest of the powers should be. So pause the video now and attempt this task. And then once you've resumed, I will be showing you what our possible answer is. Okay, so to make the largest value, I have raised negative one to negative four times negative 12 times negative two, so that's to negative 21, and that's going, to negative 24 rather. And that's going to give me the number one. Whereas for this one, I raised three to negative eight, and that gives me 1,600 and, 1000 that gives me 6,561. So this is the largest value I could possibly make from rearranging this, so that one, negative one is my base and my power is raised to negative 24. And for this one, positive three is my base and my power is negative eight. So this is the smallest value I could have gotten. What did you get then? Did you manage to put, say for example negative two as the base, what did you get when you did that? Did you put negative three as the base? What did you get when you did that? So this is this, these are the values I came up with, I wonder what you got when you did your explore task. We have now reached the end of today's lesson, a very big well done for sticking all the way through, and hopefully you have expanded your knowledge on indices. I know you now know what to do, how to evaluate and simplify negative indices. Do complete the quizzes, that would help you know if you know of how to simplify and evaluate negative indices, and I will see you at the next lesson. Bye.

Negative indices

Negative indices

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Starter quiz

8 Questions

Exit quiz

6 Questions

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UnitMaths / Indices and standard form

Maths