Video Transcript
If 𝑑 is equal to four, which of
the following is equal to the volume of this cube? Option (A) 20 to the power of six,
option (B) 20 cubed, option (C) nine to the power of 18, option (D) nine to the
power of six, or option (E) 20 to the power of 18.
In this question, we’re given the
diagram of a cube. And from this diagram, we can see
every side of our cube has length five 𝑑 all raised to the power of six. And in fact, the question tells us
the value of 𝑑. We’re told that 𝑑 is equal to
four. We need to determine the volume of
our cube. To do this, we need to start by
recalling how we find the volume of our cube. We need to recall that if a cube
has a side length we’ll call 𝑠, then we can calculate this volume as 𝑠 cubed. In other words, to find the volume
of a cube, we need to cube the length of one of its sides. And there’s several different ways
we could do this; however, we’ll only go through one of these.
Remember, we know the length of the
sides of our cube. All of the cubes have side length
five 𝑑 all raised to the power of six. But we know the value of 𝑑. In this question, we’re told that
𝑑 is equal to four. So we can substitute this to find
the length of our side. Substituting 𝑑 is equal to four
into the expression we have for the length of our side, we have the side length of
our cube 𝑠 is equal to five times four all raised to the power of six. And we can evaluate this. Inside of our parentheses, we have
five multiplied by four, and this is equal to 20. And remember, we’re raising this to
the power of six. So 𝑠 is equal to 20 to the power
of six.
But remember, this is only the
length of the sides of our cube. We need to cube this value to find
the volume. So the volume of our cube is equal
to 𝑠 cubed. And we just showed the value of 𝑠
is 20 to the power of six. So the volume is equal to 20 to the
power of six all cubed. And this is a very
complicated-looking expression. However, we can simplify this. Remember, when we cube a number, we
multiply it by itself and then multiply by itself again. So in fact, this is equal to 20 to
the power of six multiplied by 20 to the power of six multiplied by 20 to the power
of six. And in all three of these cases, we
have an integer power. This means we can simplify this by
using the product rule for monomials.
We need to recall 𝑥 to the power
of 𝑚 times 𝑥 to the power of 𝑛 is equal to 𝑥 to the power of 𝑚 plus 𝑛. In other words, to multiply these
together, all we need to do is add their powers. There’s a few different ways of
doing this. Let’s just start with simplifying
20 to the power of six multiplied by 20 to the power of six. To do this, all we need to do is
add the powers together. This is equal to 20 to the power of
six plus six. And of course, we still need to
multiply this by our other factor of 20 to the power of six. Now, we could simplify this. However, we could also notice we
can just apply our product rule again.
To multiply these together, all we
need to do is add their powers. In other words, this is just equal
to 20 to the power of six plus six plus six. And finally, we can evaluate
this. Six plus six plus six is equal to
18. So this is just equal to 20 to the
power of 18. And this was given to us as option
(E). Therefore, we were able to show if
a cube has side length five 𝑑 all raised to the power of six and 𝑑 is equal to
four, then the volume of this cube can be represented as 20 to the power of 18,
which was option (E).
