Lesson video

Hi, my name is Mr. Chang.

And in this lesson, we're going to be learning about negative integer indices and the power of zero.

Good way to understand what we're going to do in this lesson is to recap some things that we should already know.

So we're going to fill in the table, where we start with two to the power three.

That just means two repeatedly multiplied by itself three times.

Remember, the index number tells us how many times repeatedly multiplied the base number.

So two to the power three means two times two times two, which is eight.

Two to the power of two or two squared means two times two, which we know is four.

And two to power one is just two by itself, so let's ask the question, What is the connection from column to column.

So there from eight, then to four, then to two, and we're decreasing the index number.

What's actually happening is we're dividing by two each time.

And that makes sense because that means we're multiplying by one two less each time.

So if we're multiplying by one, two less, we're dividing by two.

So if we continue this pattern of the index numbers decreasing, so we start with two power three, then to the power of two, then to the power of one, it would then move on to two to the power zero, and then into the negative index numbers.

So if we continue this pattern of dividing by two, two divided by two would get us one.

Continuing again, one divided by two, another way of writing one divided by two is a half.

Divide that by two again, a half divided by two would get us one quarter.

And the quarter divided by two would get us one eighth.

Now the interesting power here and pattern today what we're learning is two to the power zero, we can see is one.

So moving on from those examples, let's have a look at these.

What is the value of each fraction and how do you know? Well, let's take each one in turn, we've got the fraction 27 to the power of eight, over 27 to the power of eight.

Now, we know that as a fraction, fractions just represent division.

So we're asking really what's 27 to the power of eight divided by 27 to the power of eight.

And I think all the other fractions are similar.

We're looking at fractions where the numerator is equal to the denominator.

So we're really dividing the value by itself.

And because we know, when you divide a fraction by itself, or divide a number by itself, is going to equal one.

And that's what we found, when we had the power two to the power zero, that equaled one.

By extending what we know about index work.

When we're dividing with powers where the base number is the same, so in all these examples, the best number is the same.

So when we're dividing with powers where the base number is the same, the base number stays the same, you subtract the powers.

So for each and every one of those, we can say that they are the base number to the power zero, so it's 27 to the power zero that must equal 1, 92 to the power zero, that equals one, nine to the power zero, that equals one, and we can see a common thing going on here.

Where all of these values to the power of zero equals one.

So that's an interesting fact that we need to try and remember that any number raised to the power zero is equal to one.

Now let's look at negative powers.

And back to the example at the beginning of the lesson, we first encountered a negative index number at two to the power of negative one.

Now how we got the value of a half was, remember we were decreasing by a power of two each time.

So that means we were dividing by the base number each time.

So to get from two to the power of zero, which we know is one now, to get two to the power of negative one, we divide by two we're dividing by the base number, so we get two to the power of negative one equals one over two, or half.

So moving forward with that principle that to get to a negative power, we simply divide by the base number, we can find these missing values.

So from 27 to the power of zero, which we know is one, to take that to 27 to the power of negative one, we would divide by 27.

That would give us one over 27.

Similarly, from 92 to the power of zero, which we know is one, to take that to 92 to the power of negative one, we would divide by 92 to give us one over 92.

And the same thing happens to go from nine to the power of zero to nine to the power of negative one, we would divide by nine to give us one of nine.

And a to the power of zero equals one, we would divide by the base number to get a to the power of negative one.

The base number in this case is just a, so we divide by a we will get one over a.

And similarly with b to power zero to it will become b to the power of negative one, we will divide by b to get one over b.

What do you notice? Well, what we notice is any number raised to the power of a negative one is equal to the reciprocal of that number.

And the reciprocal is just a number that we would multiply by to get equal one.

So let's look back at this table to help us with the negative index numbers.

Now I'm wondering whether you can spot anything.

So let's have a look.

What can you spot? So hopefully did spot a pattern, but let's have a look at some extra examples.

If we know that two to the power three equals eight, then we can see from the table two to the power negative three equals one over eight.

And that works with the all base numbers as well.

If we know that three raised power three equals 27.

So that's it three multiplied by itself three times.

Three to the power of negative three would be one over 27.

And nine squared we know is 81.

So nine to the power of negative two would equal one over 81.

And a to the power of p equals x.

What do you think a power of negative p would be? Yep, it would be one over x.

So looking at these examples, here, we can generalise a rule that we can use to help us.

So when we say a to the power of negative n, that just means the base number to any negative power.

That would be equal to, one over any number to the base to the index number.

So it will be one over a to the power of n.

Here's a question for you to try.

Pause the video to complete the task.

Resume the video once you're finished.

Here are the answers.

Now if this table asks you to continue a pattern where the index number was increasing, you would simply be multiplying by an extra base number each time.

However, this table is asking you to spot the patterns, what's happening when the index number is decreasing, and even going into the negative numbers.

So if you're working in that way, you will be dividing by the base number each time.

As you can see.

Here's some more questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

If we look at the first couple of questions here is a really good example of how a negative power is simply the reciprocal of the positive power.

So if for example, the four to the power of one is four, then four to the power of negative one is one over four.

Similarly in Part B, you know that three to the power of three is 27.

So three to the power of negative three would be one over 27.

Here are some more questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

In this question, we're asked to write each number as a power of three.

So in the form three to the power of a number.

So let's look at Part A.

In Part A, we're asked to write one over 27 as a power of three.

So what do I know about 27? Well, 27 is three cubed.

So I'm thinking that number is simply one over three cubed.

And looking at the examples that we did previously, one of the three cube is equal to three to the power of negative three.

Here's some more questions for you to try.

Pause the video to have a go, click on resume when you finished.

Here are the answers.

How did you get on? As you can see from the questions and the answers, the ideas in today's lesson, extend to algebra as well, because it doesn't really matter what base you are dealing with.

Is the idea of what happens with the index number, and particularly in today's lesson negative index numbers.

And a question I want to have a look at is question c, where we've just got simply r.

Now in terms of an index form, we can think of a number or a letter by itself just as the power of one.

We don't generally write that as a power of one.

But it's nice to understand that a number by itself is just a number to the power of one.

And that's all we've got time for in this lesson.

Thanks for watching.