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Video: Negative Powers

Tim Burnham

We will explain how and why negative powers or exponents are defined the way they are and demonstrate their use in a series of examples.

11:51

Video Transcript

In this video, we’re going to look at negative powers. Powers are sometimes called indices or exponents depending on where you live. With powers or exponents or indices, we have a base number and we have an exponent. And the exponent is telling us about how many times we have to multiply that number, the base number, by itself. So three to the power of one means we’ve just got three. Three squared, three to the power of two means we’ve got — we write down three twice, put a multiplication sign between the threes, and then we’ve got three times three. Three to the power of three is equal to three times three times three.

So we can evaluate the value of these powers. Three to the power of one is just three. Three to the power of two, three times three is nine. Three to the power of three, three times three times three is twenty-seven. Three to the power of four, three times three times three times three is eighty-one. So you should be able to see every time we add one to the power, we are multiplying by the base one more time. So three to the power of one is three. If we add one to the power to make that three to the power of two, we multiply three by three to make nine.

Let’s do the same again. Add another one to the power to make it the power of three means we’re multiplying by three. Again, nine times three is gonna give us twenty-seven. One final time, add one to the power, multiply it by three to give us eighty-one. So in general, adding one to the power is the same as multiplying by the base one more time.

So let’s take a look at the reverse of that. If I subtract one from the powers, if I take the power down from four to three by subtracting one, what do I do to eighty-one to turn it into twenty-seven? Well I do the opposite of multiplying by three. I’m dividing by three. Let’s do the same again. To reduce the power from three to two, I’m taking away one from the power. I’m dividing twenty-seven by three, the base, in order to get nine. And then one more time, every time I subtract one from the power, I’m taking my evaluated expression and dividing it by the base one more time. So subtracting one from the power is the same as dividing by the base one more time. That’s the effect we have. So let’s use that information to take this one step further.

If we started off with three to the power of one and subtracted one from that power, we’re gonna get three to the power of zero. Now what value is that gonna have when we evaluate it? Well because we take the previous expression, three, and we’re going to divide that by three, three divided by three is equal to one. So three to the power of zero is one. So if you think about that, any base to the power of one is the same value as that base. So three to the power of one is three. Five to the power of one would be five. Now if I took one away from that power, one down to zero, I’m going to divide by the base. So anything to the power of zero is just the base divided by itself, and something divided by self by itself is one. So anything to the power of zero is one. Well there is one slight exception to that. Zero to the power of zero isn’t one, but you can go and research that yourself. If you want to know what the answer of that is, that’ll be covered elsewhere.

Right, so let’s do that again. Let’s take the power of zero and subtract one from that. Well what’s the answer to three to the power of minus one? Well I’m taking this one that we had before, three to the power of zero, and I’m gonna divide that by three. Because I’ve reduced the power by one, I’m going to divide the answer by three. And one divided by three, well, the simple answer to that is one over three, a third. And let’s repeat the cycle again. Subtracting one from the power again, I’ve got three to the power of minus two.

Dividing by three, well, this is slightly more tricky. So a third divided by three, well, that’s the same as a third divided by three over one. Because three over one is the same as three. And in order to evaluate this, we say dividing fractions as easy as 𝜋. Flip the second and multiply. So that’s a third divided by three is the same as a third times a third. And one times one is one squared, which is just one. And three times three is three squared, which is nine. So three to the power of minus two is one over three squared, which is one over nine. And then one final time, subtract one from the power again to give us three to the power of negative three. And we’ve got to do a ninth divided by three. So that’s a ninth divided by three or three is the same as three over one. Dividing fractions is easy as 𝜋. Flip the second and multiply. So a ninth divided by three is the same as a ninth times a third. So we got one over nine is one over three times three. And multiply that by one over three, we’ve got one over three cubed. And one over three cubed is one over twenty-seven.

So let’s take a little look at what we’ve done there. So for example three to the power of one was three. Now three to the power of minus one is one over three. Three to the power of two was nine or three squared. So three to the power of minus two is one over three squared or one over nine. Three to the power of three was twenty-seven or three cubed. And three to the power of minus three is one over three cubed or one over twenty-seven. So there’s a pattern that we can use. That’s the definition of negative powers.

So we can summarise then. Negative powers do no reciprocals or one over, so 𝑥 to the power of negative 𝑛 is the same as one over 𝑥 to the 𝑛. Well let’s have a look at a couple of examples then. So five to the power of negative two is one over five to the power of two. So that negative sign over here told us to flip the base. And then we just keep the two here. And of course, one over five all squared is the same as one squared on the top and five squared on the bottom. And then we can evaluate that. One squared is one. Five squared is twenty-five. So our answer is one over twenty-five.

Now let’s consider ten to the power of negative five. Well firstly, we’re going to flip that base and that gives us one over ten. And because we’ve done the flipping, we can then get rid of the minus sign from the power. And that gives us one to the power of five over ten to the power of five. And now we’re going to evaluate the result. One to the power of five is one. Ten to the power of five is one hundred thousand. And that will be our answer, one over a hundred thousand.

Now let’s consider a case where we start up with a base which is a fraction, so three quarters to the power of negative two. So firstly, we’re gonna deal with a negative power by flipping the base. So flipping three over four makes four over three. So we’ve dealt with the negative sign in the power, so that’s four over three all to the power of two. And remember, four over three all to the power of two is four squared over three squared. And when we evaluate that, we get sixteen over nine.

Let’s have another example then: two-thirds all to the power of negative four. Well firstly, we’re gonna flip the base. Now that’s dealt with the negative sign in the power, so it’s three over two to the power of four. And as we saw before, three over two all to the power of four is three to the power of four over two to the power of four. And when we evaluate three to the power of four, three times three times three times three is eighty-one. Two to the power of four, two times two times two times two is sixteen, so our answer is eighty-one over sixteen.

Now you might also encounter questions like this one: zero point eight to the power of negative three. Now you could obviously do this as a big calculation: zero point eight times zero point eight times zero point eight. But quite often, there’s a simpler easier way of doing this. So let’s first of all convert that nought point eight into a fraction. So this format of the expression, eight over ten, is the same as nought point eight, is equivalent to the previous expressions. Now I’ve already used numbers one and two to describe the steps. So step zero, let’s go back from that, is to convert to a fraction, if you started off with a decimal. Now actually, we can simplify that fraction. Eight-tenths is the same as four-fifths. And now we have the question in this format. This is much like what we’ve just been looking at. So first of all, we’re going to flip the base. So we’ve dealt with the negative sign, so it becomes a positive power three. And we flipped the four over five to make it five over four. And remember, five over four all cubed is five cubed over four cubed, so we can evaluate that expression. And then we have five cubed is a hundred and twenty-five. Four cubed is sixty-four. So one two five over sixty-four is equivalent to zero point eight to the power of negative three.

So let’s just summarize that then. 𝑥 to the power of negative 𝑦 means the reciprocal of the base: one over 𝑥 to the power of 𝑦. So that negative sign in the power isn’t creating a negative answer. What it’s doing is telling us to find the reciprocal of the base. We’re dividing by the base one more time. And one over 𝑥 all to the power of 𝑦 is the same as one to the power of 𝑦 over 𝑥 to the power of 𝑦.

So I’ll just leave you with a couple of final examples: eleven to the power of negative two. First of all, we’re gonna flip the base. That negative sign in the power tells us to flip the base, so that’s one over eleven all squared. One over eleven all squared is this thing here, is one squared over eleven squared. And when we evaluate that, we get our answer one over one hundred and twenty-one. And if we have six over seven to the power of negative three, first of all, we’re gonna flip that base six over seven becomes seven over six, because we have the negative sign up here in the power. Then seven over six all to the power of three is seven to the power of three over six to the power of three, which when we evaluate that seven cubed is three hundred and forty-three. Six cubed is two hundred and sixteen. And there’s our answer.

And finally then, let’s look at a decimal example. So nought point two five to the power of negative two is where we’re gonna convert that to a fraction. So a quarter to the power of negative two, flip the quarter to make four over one, so this is gonna be four squared over one squared. And four squared is sixteen, so that would be our answer.