### 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.