### Video Transcript

Which of the following is equal to
negative two and one quarter to the third power? Option A) negative eight and one
over 64, option B) negative 11 and 25 over 64, option C) negative six and
three-quarters, option D) negative two and three-twelfths, option E) negative 27
over 12.

Let’s start this question by
writing our fraction negative two and one-quarter as a top heavy or improper
fraction. Let’s start by taking our fraction
two and a quarter ignoring the negative for a second and visualising it as two whole
shapes and one-quarter. If we split our two whole shapes
into quarters, we would have four-quarters in each one plus our one-quarter
leftover. So two and quarter is equivalent to
nine over four. So negative two and a quarter is
equivalent to negative nine over four. So finding the third power of
negative two and a quarter is equal to finding the third power of negative nine over
four.

We can recall that finding the
third power of a value 𝑥 is equivalent to working out 𝑥 times 𝑥 times 𝑥. So for negative nine over four to
the third power, we can work out negative nine over four times negative nine over
four times negative nine over four. Note that since the negative was
also part of the number taken to the third power, then it appears three times as
well. This line is equivalent to negative
nine times negative nine times negative nine over four times four times four, since
when we multiply fractions, we can multiply the numerator is together and the
denominators together. It’s important that we put the
negatives only on the numerator or only on the denominator, but not both.

To start evaluating our numerator,
we multiply a negative nine by negative nine, which is 81. Notice that we have a positive
value here, since two negatives multiplied will always give a positive value. Next, we work out 81 times negative
nine giving us negative 729. Next, we work out four times four
on the denominator giving us 16. And then we multiply this by four
which is 64. So our answer then is negative 729
over 64. And now we put answer into a mixed
number form since that is the form given in the question and in the answer
options. To do this, we need to work out how
many times 64 goes into negative 729. Our calculation in this case would
be negative 729 divided by 64 which we can do without a calculator using a long
division method.

We can work out 729 divided by 64
and then restore the negative into our answer at the end. Using the division method, we can
see that there is a whole number answer of 11 and a remainder of 25. So for our mixed number answer to
negative 729 over 64, we can start by restoring our negative. The 11 is the whole number
answer. The remainder of 25 will be the
numerator of our fraction. And we put 64 as our denominator,
since it’s the number we divided by and the number of powers in our fraction. So our answer then is the one given
an option B, negative 11 and 25 over 64.