### Video Transcript

Work out the cube root of 27
multiplied by two to the power of negative two. Give your answer as a decimal.

Let’s consider the two parts of
this product separately to begin with. We need to work out the cube root
of 27 and also two to the power of negative two. The cube root of 27 just means the
number that we multiply together — that’s by itself three times — in order to give
27.

It’s a good idea to learn the first
few cube numbers off by heart. For example, one multiplied by one
multiplied by one is one. So one is a cube number. Two multiplied by two is four and
multiplying by two again gives eight. So eight is also a cube number. Three multiplied by three is nine
and multiplying by three again gives 27. so 27 is a cube number. 27 can, therefore, be written as
three cubed, which in turn means that the cube root of 27 is equal to three. It will be sensible to learn the
next couple of cube numbers as well. Four cubed is 64, five cubed is
125, and six cubed is 216.

Now, let’s consider the second part
of this product: two to the power of negative two. And in order to work this out, we
need to recall what is meant by a negative power. A negative power defines a
reciprocal. The general rule is that if we have
𝑥 to the power of negative 𝑎, then this is equal to one over 𝑥 to the power of
𝑎.

So two to the power of negative two
is equal to one over two to the power of two or one over two squared. Two squared is equal to four. So we have that two to the power of
negative two is equal to a quarter. You can see why this definition of
negative powers works if you consider the powers of two. Two cubed is eight, two squared is
four, and two to the power of one is two.

Each time, when we decrease the
power by one, we divide by two. So eight divided by two gives four,
four divided by two gives two. Two to the power of zero then is
equal to one. To get to two to the power of
negative one, we have to divide one by two which gives a half. And then to find two to the power
of negative two, we have to divide by two again. A half divided by two is
one-quarter, which we can also write as one over two squared.

So now we’ve worked each of these
powers out individually, we can return to working out their product. We have that the cube root of 27
multiplied by two to the power of negative two is equal to three multiplied by
one-quarter. This is equal to three-quarters,
which you may be able to see straightaway. But if not, you can write the
integer three as the fraction of three over one and then multiply the two fractions
together by multiplying the numerators to give three and multiplying the
denominators to give four.

However, the question asked us to
give our answer as a decimal. So finally, we need to convert
three-quarters into a decimal. This is equal to 0.75. Now, this is one of the common
decimal fraction equivalents that you should know off by heart. But if you didn’t know that
three-quarters was equal to 0.75, you could work this out using a short
division.

We set up our short division with
three underneath the bus stop division sign and four on the outside. We’ll include the decimal point and
add some extra zeros after it. There are no fours in three. So we put a zero and carry the
three into the next column. Seven fours are 28. So there are seven fours in 30 with
a remainder of two. Five fours are 20 with no
remainder. So we found by division that three
divided by four is equal to 0.75.

We’ve worked out then that the cube
root of 27 multiplied by two to the power of negative two as a decimal is equal to
0.75.