Video: Zero and Negative Exponents

In this video, we will learn how to simplify and evaluate expressions involving zero and negative indices.

17:57

Video Transcript

In today’s lesson, what we’re looking at is zero and negative exponents. Well, it’s worth knowing that exponents can also be called powers or indices depending on whereabouts you are. And when we’re looking at exponents, powers, or indices, we could see that these are, in fact, a small number we’d have above another number we’d call the base.

So, as you can see here, we’ve got our three as our base and our zero as our exponent. Well, negative exponents and zero exponents, what do they actually mean? Well, let’s take a look. Let’s see what exponents are and how we could find out what zero and negative exponents also are.

Well, first, let’s recap what an exponent or a power is? Well, if we got three raised to the exponent four or raised to the power of four, then what this is gonna mean is three multiplied by three multiplied by three multiplied by three. Remembering that, in this example here, the three would be the base number and the four would be the exponent or power or indices. But we’ve got here that this means three multiplied by three multiplied by three multiplied by three. So, what the exponent is telling us is how many times, in fact, we are multiplying the base by itself. Because we can see that we’ve got four threes and then we’ve got a multiply sign in between each of these.

It’s worth noting that our multiply sign, we’ve got a dot here. But in some regions, you’ll also see this might be a cross. Either of these can be used, a bit like how we might use exponent, power, or indices. Okay, so then, we’ve got three cubed. This is three multiplied by three multiplied by three. Three squared is three multiplied by three. And three to the power of one is just three.

Well, let’s evaluate these and take a look at what the actual value of these are. So, we’ve got three to the power of four is 81. Three cubed is 27. Three squared is nine. And three to the power of one, as we’ve said, is just three. Well, let’s see how this all works. Well, we could see here that every time we add one to our exponent, then what this means is that you can multiply by the base one more time. Because if we look up, we’ve gone from three to nine, then nine to 27, then 27 to 81. So, each time, we’re multiplying by three.

So, great, we’ve looked at what exponents are and how you work them out. But we need to think what’s this lesson about. It’s about zero and negative exponents. So, let’s have a little look about how these might work. Well, first of all, we’re gonna take a look at three raised to the power of zero. Well, we don’t know what this is, but could we work it out using the pattern that we’ve got?

Well, if we work backwards, we can see that, in fact, every time we subtract one from the exponent, then what we do is we divide by three. Cause 81 divided by three is 27, 27 divided by three is nine, nine divided by three is three. So therefore, if we divide three by three, we’re just left with one. So, we can say that three to the power of zero is equal to one. And in fact, anything raised to the power of zero is just equal to one. So, great, we worked out our zero exponent, what about negative exponents?

Well, with the negative exponent, so we’re gonna go down from zero to negative one, what we need to do is divide by three once more. So, if we have one divided by three, we’re gonna get one-third. So, we’re gonna say that three to the power of negative one is equal to a third. But can we get a general form from this? Well, let’s see if we can do it one more time. Well, if we go down to negative two, so three to the power of negative two, then we’re gonna divide by three once more. But this isn’t so straightforward because a third divided by three, how are we going to do that?

Well, I’m gonna show the calculation cause it help us see what the result is. So, if we’ve got a third divided by three, we can also think of it as a third divided by three over one. Well, if we’re dividing by a fraction, then what we do is we find the reciprocal. So, we flip that fraction. That’s the second fraction. And then, we multiply. So, this is gonna give us one-third multiplied by one-third, which is gonna give us one squared over three squared, which will be equal to one over nine. So, great, from this we can now think about what our general form is gonna be.

So, the general form is 𝑥 to the power of 𝑛 is equal to one over 𝑥 to the power of 𝑛. So, we’re gonna see that this is, in fact, a reciprocal. Okay, great, well, let’s take a look at how this might work in practice. Let’s have a look at a question.

Which of the following is equal to 46 to the power of negative one? (A) One over 46, (B) 46, (C) 45, (D) one over 45, or (E) 0.46.

Well, what we can recall is that we’ve got 𝑥 to the power of negative 𝑛 equals one over 𝑥 to the power of 𝑛. So, this is one of the rules that we have of our exponents. So, from this, we can see that 𝑥 to the power of negative one is gonna be equal to one over 𝑥. So therefore, this, in fact, is gonna be the reciprocal. So, let’s take a look at what we’ve got. We’ve got 46 to the power of negative one. Well, if we think of 46 as 46 over one, then 46 to the power of negative one is gonna be the reciprocal of 46 over one. Which is gonna give us one over 46 because what we do is we flip the numerator and the denominator to find our reciprocal.

So therefore, we can say that answer (A) is equal to 46 to the power of negative one. And that’s because answer (A) is one over 46.

Okay, so, that was a nice example just to sort of show us what we would get if we had a power of negative one. But where are we gonna move on to next? Well, let’s take a look at the example five to the power of negative two. Well, once again, what we do is we recall 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. So therefore, five to the power of negative two is gonna be equal to one over five squared, which would give us one over 25.

It is worth noting though, as we showed earlier, that, in fact, what we have is one squared over five squared, which would give us one over 25. Just in our general form, we put 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. But this will come in very useful in later on with questions that we’re gonna do.

So, what now? Well, now, it’s time for the next question. So, let’s move on and see what we’ve got next. Well, in this question, what we’re gonna take a look at is one that involves a decimal, so this can be quite interesting. So, let’s get it on and solve the problem.

What is the value of 0.2 all to the power of negative three?

Well, if we think about 0.2, what the 0.2 is is the same as two-tenths. And that’s because if we look at the place values, our zero is the units and our two is the tenths. So, we’re gonna say that 0.2 is equal to two-tenths. But can we make this any simpler? Well, yes, we can simplify the fraction by dividing the numerator and denominator by two. And when we do this, what we get is one over five or one-fifth because two divided by two is one and 10 divided by two is five. So, we know that 0.2 is equal to a fifth. And in fact, this is a decimal conversion that we should know.

So, what we’re gonna get is one-fifth to the power of negative three. Well, if we recall that we got 𝑥 to the power of negative 𝑛 equals one over 𝑥 to the power of 𝑛, it’s slightly different here because we’ve got a fraction, one over five. But it’s exactly the same theory because what we’re going to do is we’re going to find the reciprocal of one over five. And then we’re gonna put it to the power of three.

So, we can think of it as five cubed over one cubed. And that’s because the reciprocal of one over five is five over one. But usually, we wouldn’t bother writing the one cubed on the denominator. Well, five cubed means five multiplied by five multiplied by five. Well, five multiplied by five is 25, and 25 multiplied by five is 125. So therefore, we can say that the value of 0.2 to the power of negative three is 125.

Okay, great, so we’ve looked at another example. And this one has a decimal in it. So, we’re now starting to get to grips with different types of negative exponents. But what’s the next type of question we can look at? Well, in this question, what we’re gonna look at is seeing if we can calculate using our negative exponents. And the answer is yes, we can.

Calculate two to the power of negative two multiplied by three to the power of negative one all to the power of negative one, giving your answer in its simplest form.

So, first of all, what we’re doing is we’re gonna use our rule that we have — that’s 𝑥 to the power of negative 𝑛 equals one over 𝑥 to the power of 𝑛 — to convert our negative exponents into factions. So, first of all, we’re gonna have two to the power of negative two. And this’s gonna be equal to one over two squared, which is gonna be equal to one over four. So, we can say that two to the power of negative two is equal to a quarter. Then, next, what we have is three to the power of negative one. And we know that three to the negative one is gonna be equal to one over three. And that’s because it’s the reciprocal of three over one. So, three over one is what three is. Reciprocal of that is one over three. Okay, great, we’ve converted them both into fractions.

Okay, great, so, now, what we have is a quarter multiplied by a third all to the power of negative one. Well, if we multiply a fraction, what we do is we multiply the numerators then multiply the denominators. So, one multiplied by one is one. Four multiplied by three is 12. So, we’ve got one over 12 or one twelfth to the power of negative one. Well, as we said before, when we’ve got something raised to the power of negative one, what we need to do is just find the reciprocal. And to find the reciprocal, what you do is you flip the numerator and denominator.

So therefore, our two to the power of negative two multiplied by three to the power of negative one all to the power of negative one becomes 12. And that’s because the final step was to have one over 12, or one twelfth, to the power of negative one. Well, the reciprocal of one over 12 is just 12.

Okay, so great, we’ve looked at calculating using our negative exponents, but now what we’re gonna do is take a look at a fraction. And this fraction’s gonna be a fraction where we’ve got a number that isn’t one as the numerator and a number that isn’t one as the denominator.

Which of the following is equal to two-thirds to the power of negative three? The options are (A) 27 over eight, (B) eight over 27, (C) negative eight over 27, (D) negative six over negative nine, or (E) negative 27 over eight.

So, to solve this problem, what we’re gonna do is we’re going to use an adaptation of the most common exponent rule. And that exponent rule is 𝑥 to the power of negative 𝑛 is equal to one over 𝑥 to the power of 𝑛. Because, in fact, what this really means is 𝑥 to the power of negative 𝑛 is equal to one to the power of 𝑛 over 𝑥 to the power of 𝑛. However, as one to the power of 𝑛 is just one, we write one over 𝑥 to the power of 𝑛. So therefore, if we’ve got 𝑥 over 𝑦 to the power of negative 𝑛, this is gonna be equal to 𝑦 to the power of 𝑛 over 𝑥 to the power of 𝑛. Cause what we do is we find the reciprocal of our fraction and we put each of the terms raised to the power of 𝑛.

So therefore, if we got two-thirds, or two over three, to the power of negative three, well, first of all, what we do is we find the reciprocal. And the reciprocal is what we get if we flip the numerator and denominator. So, two-thirds becomes three-halves or three over two. Well, then, what we’re gonna do is raise both the numerator and the denominator to the power of three because, in our example, this is our 𝑛. Well, three cubed means three multiplied by three multiplied by three, and two cubed means two multiplied by two multiplied by two.

So therefore, we can say that two-thirds to the power of negative three is gonna be equal to 27 over eight. And that’s because three times three times three is 27 and two times two times two is eight. So therefore, we can say that the correct answer from our list is going to be answer (A) because this is 27 over eight or twenty-seven eighths.

So, great, what we’re gonna do now is move on to the final leg of our journey with negative exponents. We started off with exponents of negative one. Then, next, what we did is we moved on to exponents that were not just negative one, negative two, negative three, et cetera. Then, next, we multiplied values that were raised to negative exponents, so for example, two to the power of negative two multiplied by three to the power of negative one. And then, we’ve just dealt with fractions, so, for example, 𝑥 over 𝑦 to the power of negative 𝑛 or two-thirds to the power of negative three.

So, where we’re looking to go next is the final leg of our journey. And we’re gonna bring together the skills that we’ve learned to multiply fractions that are raised to negative exponents.

Which of the following is equal to negative three-quarters to the power of five multiplied by negative three-quarters to the power of negative seven. The options are (A) one and seven-ninths, (B) negative three-quarters to the power of negative 35, (C) negative one and seven-ninths, (D) nine over 16, or (E) negative nine over 16.

So, in order to solve this problem, what we’re gonna do is look at a couple of general rules we have for exponents. First of all, if we have 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏, this is equal to 𝑥 to the power of 𝑎 plus 𝑏. So, we add the exponents. And for the second rule, we’ve got 𝑥 over 𝑦 to the power of negative 𝑛 is equal to 𝑦 to the power of 𝑛 over 𝑥 to the power of 𝑛. So, what we do is we find the reciprocal of our fraction and then we put both the numerator and denominator to the power of 𝑛.

And we can use our first rule because we notice in our question that we, in fact, have both of our bases are the same because they’re both negative three over four or negative three-quarters. So therefore, negative three-quarters to the power of five multiplied by negative three-quarters to the power of negative seven is gonna be equal to negative three-quarters to the power of five add negative seven. Which is gonna be equal to negative three-quarters to the power of negative two.

So, now, what we’re gonna do is move on and apply the second rule. So, to apply the second rule, what we do, first of all, is find the reciprocal. So, instead of three over four, we’re gonna have four over three. But because we had negative three over four, what we’re now gonna have is negative four over three. But what I’m gonna do is I’m gonna put the negative with the numerator just so it doesn’t get left out or just because we don’t get the wrong answer, i.e., negative when we should get positive, et cetera.

So, then, we’ve got negative four squared over three squared. And that’s because they’re both to the power of two. And this is gonna give us 16 over nine. And that’s because negative four multiplied by negative four is 16 and three multiplied by three is nine. And then, to change this it into a mixed number, what we do is we see how many nines go into 16. And they go into 16 once remainder seven. And we do that because if we looked at our answers (A), (B), (C), (D), and (E), none of them are 16 over nine.

So therefore, our answer is gonna be one and seven-ninths. So, now, we can take a look at our answers on the left-hand side. So therefore, we can match this to answer (A). And we see that answer (A) must be the correct answer. So, (A), which is one and seven-ninths, is the correct answer. But also, it’s worth noting that we could’ve got some of the other answers with some simple mistakes.

For example, we could’ve got answer (C) if we’d forgotten about the negative sign that we’d mentioned earlier. Because if we’d forgotten about the negative sign and just left it outside of our fraction, then what we would have had is negative 16 over nine, which would have given us negative one and seven-ninths.

And another common mistake that could have brought us answer (B) would have been one where we multiplied the exponents instead of adding them. So, we wouldn’t use the first rule. We’d use the rule that was incorrect that said if you multiply numbers that are the same base and different exponents, then you multiply the exponents. And that would’ve given us negative 35. So, we’ve avoided that. And we’ve got the correct answer, which is one and seven-ninths or answer (A).

So, fantastic, what we’ve done now is we finished our journey. And we’ve reached the end. So, now, all we need to do is recap the key points. Well, the key points we’re learning today are that if you have three to the power of two, then three would be the base and two would be the exponent or power. If we have 𝑥 raised to the power of zero, this is just equal to one. If we had 𝑥 to the power of negative 𝑛, then this is equal to one over 𝑥 to the power of 𝑛. So, it’s the reciprocal. And then, finally, if we have 𝑥 over 𝑦 raised to the power of negative 𝑛, this is equal to 𝑦 to the power of 𝑛 over 𝑥 to the power of 𝑛. So, we find the reciprocal of our fraction and then raise both the numerator and the denominator to the power 𝑛.

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