### Video Transcript

In this video, we will learn how to
find cube roots of perfect cube integers. We begin by recalling that we can
use the square root to determine the side length of a square from its area. For example, given that a square
has an area of 16 square centimeters and its side length is called π centimeters as
shown, then we know that π squared is equal to 16. We can then solve for π by taking
the square root of both sides of the equation, noting that since π is the length,
it must be positive. We have π is equal to the square
root of 16, which is the same as the square root of four squared. And we can therefore conclude that
π is equal to four. The side length of the square is
four centimeters.

We can extend this process to
cubes. Letβs assume that we have a cube of
volume 125 cubic centimeters. Recalling that the volume of a cube
is the cube of its side length, we have π cubed is equal to 125. This time, we can cube root both
sides of our equation, giving us π is equal to the cube root of 125. We know that four cubed is equal to
64, so the cube root of 64 is four. Five cubed is equal to 125. This means that the cube root of
125 is five, and the cube has side length five centimeters. It is worth noting at this stage
that cube rooting a positive number gives a positive answer, whereas cube rooting a
negative number gives a negative answer. Since 125 is the cube of an
integer, it is known as a perfect cube. Other examples of perfect cubes are
one, eight, 27, and 64.

Before looking at some specific
examples, we will define this more formally. A perfect cube is an integer that
is equal to the product of the same integer three times. For example, eight is a perfect
cube, since eight is equal to two multiplied by two multiplied by two. We can also say that an integer π
is a perfect cube if there is an integer π such that π cubed is equal to π. We can also define the cube root of
a number in the same way we define the square root of a number. The cube root of a number π,
written as shown, is the reverse operation of cubing a number. In general, the cube root of π is
the number π such that π cubed is equal to π. Letβs now see an example where we
need to work out the cube root of a perfect cube.

Evaluate the cube root of 27.

We begin by recalling that the cube
root of a number π, written as shown, is the number π such that π cubed is equal
to π. In this question, we are asked to
find the cube root of 27. We know that three cubed is equal
to three multiplied by three multiplied by three. And this is equal to 27. We can therefore conclude that the
cube root of 27 is equal to three.

It is worth noting that the number
27 is a perfect cube, since its cube root is an integer. We also note that cube rooting a
positive number gives a positive answer, whereas cube rooting a negative number
gives a negative one.

Letβs now consider this in more
detail. We begin by noting that, unlike the
square root, the cube root always exists and gives a unique solution. For example, we recall that there
are two square roots of four, since two squared is equal to four and negative two
squared equals four. The square root of four is equal to
two or negative two. There is only one cube root of
eight, since only two cubed equals eight. We also note that negative two
cubed is equal to negative eight, and so the cube root of negative eight is negative
two. This confirms that we can take the
cube root of a negative number and our answer itself will be negative.

In our next example, we will
determine the cube root of a negative perfect cube.

Find the value of the cube root of
negative one.

We begin by recalling that the cube
root of a negative number is negative. We also recall that the cube root
of a number π is the number π such that π cubed is equal to π. This means that in this question,
we need to find a negative number that when cubed gives negative one. The only solution here is negative
one, since negative one cubed is equal to negative one multiplied by negative one
multiplied by negative one, which in turn is equal to negative one. We can therefore conclude that the
cube root of negative one is negative one.

Before moving on to our next
example, there is another way to determine the cube root of a perfect cube. Instead of using trial and error to
find the cube root, we can use prime factorization to simplify the process. Letβs consider the cube root of
3375 to demonstrate this. We begin by noting that 3375 is
divisible by five. So this is a factor. 3375 divided by five is 675, and
this is also divisible by five, giving us 135.

We can divide by five once more,
giving us 27. And noting that 27 is equal to
three cubed, then 3375 must be equal to five cubed multiplied by three cubed. We can therefore rewrite the
original calculation as shown. And using our laws of indices or
exponents, five cubed multiplied by three cubed can be rewritten as five multiplied
by three all cubed. And our expression simplifies to
the cube root of 15 cubed. Since the cube root of π cubed is
equal to π, then the cube root of 15 cubed is 15. And we can therefore conclude that
the cube root of 3375 is 15 and that 3375 is a perfect cube.

This can be summarized in general
as follows. If we have a number that is the
product of perfect cubes π, which is equal to π multiplied by π where π and π
are perfect cubes β for example, π is equal to π cubed and π is equal to π cubed
β then since ππ all cubed is equal to π cubed multiplied by π cubed, this is
equal to π multiplied by π, which is equal to π. And hence, the cube root of π is
equal to the cube root of ππ, which is equal to ππ. This allows us to take the cube
roots of the products of the same prime factors separately. For example, the cube root of five
cubed multiplied by three cubed can be rewritten as the cube root of five cubed
multiplied by the cube root of three cubed, which in turn is equal to five
multiplied by three, which is equal to 15.

In our next example, we will
simplify an expression involving both square and cube roots.

Find the value of the square root
of negative 55 multiplied by the cube root of negative 216.

In this question, weβre given an
expression that includes a cube root inside of a square root. We will begin with the inner
expression, that is, working out the value of the cube root of negative 216. We recall that the cube root of a
number π is the number π such that π cubed is equal to π. And since negative 216 is negative,
we need to find a negative number that when cubed gives us negative 216. We know that 216 is a perfect cube,
since six cubed is equal to 216. Using the properties of perfect
cubes, this means that negative six cubed is equal to negative 216. And as such, the cube root of
negative 216 is negative six.

Substituting this into the
expression, we have the square root of negative 55 multiplied by negative six. Recalling that multiplying two
negative numbers gives a positive answer, and since 55 multiplied by six is 330,
then negative 55 multiplied by negative six is also 330. And our expression simplifies to
the square root of 330. We could try to evaluate this
further. However, on inspection, we note
that 330 is not a perfect square. And hence, the square root of
negative 55 multiplied by the cube root of negative 216 is the square root of
330.

Before looking at one final
example, we will formally define a useful property of the cube root function.

When finding the cube root of a
perfect cube for any integer π, the cube root of π cubed is equal to π. As we have already seen in this
video, if we wish to calculate the cube root of 64, then since 64 is a perfect cube,
we can rewrite this as four cubed, giving us the expression the cube root of four
cubed, which is equal to four. Letβs now look at an example where
we can use this property in context.

What is the edge length of a cube
whose volume is 64 cubic centimeters?

We begin by recalling that the
volume of a cube with an edge length of π centimeters is given by the cube of
π. In this question, since the volume
of the cube is 64 cubic centimeters, then π cubed equals 64. Taking the cube root of both sides
of this equation, we have π is equal to the cube root of 64. We know that 64 is a perfect cube,
since it is equal to four cubed. And π is therefore equal to the
cube root of four cubed. Next, for any integer π, we know
that the cube root of π cubed is π. And this means that π must be
equal to four. We can therefore conclude that the
edge length of a cube whose volume is 64 cubic centimeters is four centimeters.

We will now summarize the key
points from this video. We saw in this video that we call
an integer π a perfect cube if there is an integer π such that π cubed is equal
to π. We also saw that the cube root of a
number π, written as shown, is the number π such that π cubed is equal to π. We can evaluate the cube root of a
perfect cube by trial and error using a calculator or by using the prime
factorization method.

The cube root function preserves
the sign of a number. This means that the cube root of a
positive number is positive and the cube root of a negative number is negative. We saw the key property that for
any integer π, the cube root of π cubed is equal to π. Finally, we saw at the start and
end of this video that we can determine the side length of a cube from its volume by
taking the cube root of the volume.