Video Transcript
Evaluate four to the power of three
over two multiplied by 32 to the power of negative 0.2 multiplied by two to the
power of negative three all over 32 to the power of negative 0.4 multiplied by two
to the power of negative two.
So to solve a problem like this,
what we need to do is we use our exponent rules. But let’s remind ourself of
them. So firstly, we have 𝑥 to the power
of 𝑎 multiplied by 𝑥 to the power of 𝑏, which is equal to 𝑥 to the power of 𝑎
plus 𝑏, so we have the exponents. Then we have 𝑥 to the power of 𝑎
divided by 𝑥 to the power of 𝑏 which is equal to 𝑥 to the power of 𝑎 minus 𝑏,
𝑥 to the power of 𝑎 all to the power of 𝑏 which is equal to 𝑥 to the power of
𝑎𝑏. Then we have 𝑥 to the power of
negative one which is equal to the reciprocal or one over 𝑥. And finally, we’ve got 𝑥 to the
power of a half which is equal to the square root of 𝑥.
There are other exponent rules that
are adaptations of these which we may use further on in the question. Well, we can notice that throughout
the exponent rules when we’re involving multiplying or dividing, et cetera, then
what happens is the base number is the same. But if we look at our expression,
the base number isn’t the same for each of the terms because we’ve got four, 32, and
32 here. But in fact we can rewrite our
expression. And we can do that with everything
having the same base. And that’s because using two, we
can actually make 32 and four. And that’s because two squared is
equal to four and two to the power of five is equal to 32. So whenever we see this type of
question, be careful and look out for this because this will generally be a way of
solving it.
Also, if we take a look at our
expression, we can see that we’ve got a couple of decimal exponents and we don’t
want this. We want to convert these into
fractions because this will make life a lot easier. Well, we know that 0.2 is equal to
a fifth, so we can change that. And 0.4 is equal to two-fifths, so
we can change that. Okay, great! So now what we can do is start to
imply our exponent rules. So the first thing we’re gonna do
is we’re going to apply the third exponent rule that we introduced you to.
Well, our first term is gonna be
two cubed. And that’s because we had two
squared all to the power of three over two. Well, if we multiply two by three
over two, we just get three. Then the next term is two to the
power of negative one. And that’s because if we have five
multiplied by negative a fifth, that’s gonna give us negative one. Then the next term remains
unchanged. Then we have two to the power of
negative two, and that’s because five multiplied by negative two over five, the
fives cancel, so we’re left with negative two. And then the final term remains
unchanged. Okay, great! So now, what’s the next step?
Well, next, what we’re gonna do is
apply the first exponent rule. So when we do that, what we gonna
have is two to the power of three plus negative one plus negative three over two to
the power of negative two plus negative two, which is gonna leave us with two to the
power of negative one over two to the power of negative four. So at this point, we might think,
“Hold on! We’ve got negative powers
here.” So we think, well, the fourth
exponent rule we looked at, that involves negative powers or exponents. So is it this that we’re going to
use? But no, we don’t need to because
what we can do is go straight to our second rule. And that’s because two to the power
of negative one over two to the power of negative four means two to the power of
negative one divided by two to the power of negative four.
So this’s gonna give us two to the
power of negative one minus negative four. Well, if we subtract a negative,
it’s the same as adding a positive. So this is just gonna give us two
cubed. But if we did want to think, well,
how could we work out using negative exponents? But we’d have two to the power of
negative one is the same as one over two or a half then divided by two to the power
of negative four, which is the same as one over two to the power of four.
Well, then, if we applied our rule
for dividing fractions, which is, keep it, change it, flip it, we’d have a half
multiplied by two to the power of four over one, which would give us two to the
power of three. And that’s because we’d have one
multiplied by two to the power of four, which is two to the power of four. Two multiplied by one is two. Or two to the power of four divided
by two is the same as two to the power of four divided by two to the power of
one. So we get two to the power of three
or two cubed.
And as two cubed is equal to eight,
then, therefore, we can say that if we evaluate our expression, which was four to
the power of three over two multiplied by 32 to the power of negative 0.2 multiplied
by two to the power of negative three all over 32 to the power of negative 0.4
multiplied by two to the power of negative two, we just get the answer eight.
