Lesson Video: Cube Roots of Perfect Cubes Mathematics • 8th Grade

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 𝑙 centimeters as shown, then we know that 𝑙 squared is equal to 16. We can then solve for 𝑙 by taking the square root of both sides of the equation, noting that since 𝑙 is the length, it must be positive. We have 𝑙 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 𝑙 is equal to four. The side length of the square is four centimeters.

We can extend this process to cubes. Let’s 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 𝑙 cubed is equal to 125. This time, we can cube root both sides of our equation, giving us 𝑙 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 𝑛 is a perfect cube if there is an integer 𝑎 such that 𝑎 cubed is equal to 𝑛. 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 𝑛, written as shown, is the reverse operation of cubing a number. In general, the cube root of 𝑛 is the number 𝑎 such that 𝑎 cubed is equal to 𝑛. Let’s 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 𝑛, written as shown, is the number 𝑎 such that 𝑎 cubed is equal to 𝑛. 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.

Let’s 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 𝑛 is the number 𝑎 such that 𝑎 cubed is equal to 𝑛. 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. Let’s 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 𝑎 cubed is equal to 𝑎, 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 𝑐, which is equal to 𝑎 multiplied by 𝑏 where 𝑎 and 𝑏 are perfect cubes — for example, 𝑎 is equal to 𝑛 cubed and 𝑏 is equal to 𝑚 cubed — then since 𝑛𝑚 all cubed is equal to 𝑛 cubed multiplied by 𝑚 cubed, this is equal to 𝑎 multiplied by 𝑏, which is equal to 𝑐. And hence, the cube root of 𝑐 is equal to the cube root of 𝑎𝑏, which is equal to 𝑛𝑚. 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, we’re 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 𝑛 is the number 𝑎 such that 𝑎 cubed is equal to 𝑛. 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 𝑎, the cube root of 𝑎 cubed is equal to 𝑎. 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. Let’s 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 𝑙 centimeters is given by the cube of 𝑙. In this question, since the volume of the cube is 64 cubic centimeters, then 𝑙 cubed equals 64. Taking the cube root of both sides of this equation, we have 𝑙 is equal to the cube root of 64. We know that 64 is a perfect cube, since it is equal to four cubed. And 𝑙 is therefore equal to the cube root of four cubed. Next, for any integer 𝑎, we know that the cube root of 𝑎 cubed is 𝑎. And this means that 𝑙 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 𝑛 a perfect cube if there is an integer 𝑎 such that 𝑎 cubed is equal to 𝑛. We also saw that the cube root of a number 𝑛, written as shown, is the number 𝑎 such that 𝑎 cubed is equal to 𝑛. 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 𝑎, the cube root of 𝑎 cubed is equal to 𝑎. 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.