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

In this explainer, 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 cm2, and its side length is called 𝑙 cm, as shown, then, we know that 𝑙=16.

Then, we can solve for 𝑙 by taking the square root of both sides of the equation, where we note that 𝑙 is a length and so it is positive. We have 𝑙=√16=√4=4.

Another way of thinking about this process is to consider that we need a number multiplied by itself to give 16. We can then try different values until we see that 4Γ—4=16.

In either case, we can extend this process to cubes. We recall that the volume of a cube is the cube of its side length. Hence, if we have a cube with volume 125 cm3 and side length 𝑙 cm, as shown, then we know that 𝑙=125.

We can once again find this value of 𝑙 by trying values. For example, we see that if we try 𝑙=4, we get 4Γ—4Γ—4=64.

This is less than 125, so we need to increase the length. We can try 𝑙=5 to get 5Γ—5Γ—5=125.

Therefore, 𝑙=5. In fact, we know this is the only solution since increasing the side length of the cube must increase its volume and decreasing its side length must decrease its volume.

We call a number like 125 a perfect cube, since it is the cube of an integer. In this case, 125=5. We can define this formally as follows.

Definition: Perfect Cubes

A perfect cube is an integer that is equal to the product of the same integer three times. For example, 8 is a perfect cube since 8=2Γ—2Γ—2.

We can also say that an integer 𝑛 is a perfect cube if there is an integer π‘Ž such that π‘Ž=π‘›οŠ©.

We can also define the cube root of a number in the same way we defined the square root of a number.

Definition: Cube Roots of Perfect Cubes

The cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the reverse operation of cubing a number. In general, the cube root of 𝑛 is the number π‘Ž such that π‘Ž=π‘›οŠ©.

Above, we showed that √125=5, since 5=125. In fact, for any perfect cube 𝑛, we know that 𝑛=π‘ŽοŠ© for some integer π‘Ž. Hence, οŽ’βˆšπ‘›=π‘Ž. We could have also used the cube root button on a calculator to find the side length of a cube given its volume. We note that 𝑙=125.

We then take the cube root of both sides of the equation to get 𝑙=√125=5.

Let’s now see an example of determining the cube root of a perfect cube.

Example 1: Finding the Cube Root of a Perfect Cube

Evaluate √27.

Answer

We recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©. So, we are looking for a number whose cube is 27. We can check a few numbers and note that 3=3Γ—3Γ—3=27.

Hence, √27=3.

A useful point worth noting is that, unlike the square root, the cube root always exists and gives a unique value. For example, we can recall that there are two square roots of 4, since 2=4 and (βˆ’2)=4. However, there is only one cube root of 8, since only 2=8.

We can also note that (βˆ’2)=βˆ’8 and so οŽ’βˆšβˆ’8=βˆ’2. This means that we can take the cube root of negative numbers, and since the cube of a negative number is negative, the cube root preserves the sign of the number. Finally, we can also note that 0=0, so √0=0.

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

Example 2: Finding the Cube Root of a Perfect Negative Cube

Find the value of οŽ’βˆšβˆ’1.

Answer

We recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©. So, we are looking for a number whose cube is βˆ’1. Since this number is negative, we will need to look for a negative number that when cubed gives βˆ’1.

We can check a few negative numbers by cubing them starting at βˆ’1: (βˆ’1)=(βˆ’1)Γ—(βˆ’1)Γ—(βˆ’1)=βˆ’1.

Hence, οŽ’βˆšβˆ’1=βˆ’1.

Before we move on to our next example, there is another way to determine the cube root of a perfect cube. Instead of trying many numbers to find the cube root, we can use prime factorization to simplify the process. Let’s find √3375 to see an example of how to do this.

First, we factor 3β€Žβ€‰β€Ž375 by noting that it is divisible by 5. We want to keep removing factors of 5 until we cannot do this anymore. We have

We then have another factor of 5

Then, one final factor of 5

Thus, 3375=5Γ—27. We then note that 27=3. Hence, 3375=3Γ—5=(3Γ—5)=15.

Since 3375=15, we must have that √3375=15.

We can verify this by checking that 15=3375 or by using a calculator.

This process works in general, since if we have a number that is the product of perfect cubes 𝑐=π‘Žπ‘, where π‘Ž and 𝑏 are perfect cubes, say π‘Ž=π‘›οŠ© and 𝑏=π‘šοŠ©, then we can note that (π‘›π‘š)=(π‘›π‘š)Γ—(π‘›π‘š)Γ—(π‘›π‘š)=π‘›Γ—π‘š=π‘ŽΓ—π‘=𝑐.

Hence, οŽ’οŽ’βˆšπ‘=βˆšπ‘Žπ‘=π‘›π‘š.

This allows us to take the cube roots of the products of the same prime factors separately. For example, consider the number 216. We can factor 216 into primes to see that 216=2Γ—2Γ—2Γ—3Γ—3Γ—3=2Γ—3.

So, the prime numbers 𝑛 and π‘š are 𝑛=2 and π‘š=3 and we have √216=√2Γ—3=√2Γ—βˆš3=2Γ—3=6.

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

Example 3: Evaluating Numerical Expressions Involving Square and Cubic Roots

Find the value of ο„βˆ’55βˆšβˆ’216.

Answer

We need to find the value of a given expression that includes a cube root inside of a square root. We should start with the inner operation since this is evaluated first. We want to determine the value of οŽ’βˆšβˆ’216.

We recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©. Since βˆ’216 is negative, we will need to look for a negative number that when cubed gives βˆ’216. We can then find that (βˆ’6)=βˆ’216. Hence, οŽ’βˆšβˆ’216=βˆ’6.

We can now substitute this into the expression ο„βˆ’55βˆšβˆ’216=βˆšβˆ’55Γ—(βˆ’6)=√330.

We could try to evaluate this further; however, on inspection, we note that 330 is not a perfect square.

Hence, ο„βˆ’55βˆšβˆ’216=√330.

In our next example, we will compare the sizes of a square and a cube root without calculating the exact values.

Example 4: Estimation of Cube Roots to Compare Two Numbers

Without using a calculator, determine if √121 is greater than, equal to, or less than οŽ’βˆšβˆ’42.

Answer

We could answer this question by first calculating the value of √121. However, this is not necessary. First, we recall that the square root of a positive number is positive and the cube root of a negative number is negative. Therefore, √121>0 and οŽ’βˆšβˆ’42<0.

Hence, √121 is greater than οŽ’βˆšβˆ’42.

In our next example, we will show a useful property of the cube root.

Example 5: Evaluating the Cube Root of a Cube

Evaluate √2.

Answer

We recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©. Thus, we want to find the number whose cube is equal to 2. In this case, 𝑛=2, so π‘Ž=2. Hence, √2=2.

In the previous example, we showed the following useful property of the cube root function.

Property: Cube Root of a Perfect Cube

For any integer π‘Ž, οŽ’βˆšπ‘Ž=π‘Ž.

This tells us that the cube root reverses the action of cubing. This is a useful result, and we can note that this is not true for the square root. Since the square root of a negative number squared does not return the original value. Instead, it returns the positive root.

In our final example, we will determine the length of the edge of a cube by using the cube root.

Example 6: Finding the Edge Length of a Cube given Its Volume Using Cube Roots

What is the edge length of a cube whose volume is 64 cm3?

Answer

We recall that the volume of a cube with an edge length of 𝑙 cm is given by the cube of 𝑙. Hence, 𝑙=64.

We can find the value of 𝑙 by considering the cube root of 64, where we recall that the cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©. In this case, we can note that 4=4Γ—4Γ—4=64.

Hence, the cube has edges of length 4 cm.

Let’s finish by recapping some of the important points from this explainer.

Key Points

  • We call an integer 𝑛 a perfect cube if there is an integer π‘Ž such that π‘Ž=π‘›οŠ©.
  • The cube root of a number 𝑛, written οŽ’βˆšπ‘›, is the number π‘Ž such that π‘Ž=π‘›οŠ©.
  • We can evaluate the cube root of a perfect cube by using a calculator or by using the prime factorization method.
  • The cube root preserves the sign of the number.
  • For any integer π‘Ž, οŽ’βˆšπ‘Ž=π‘ŽοŠ©.
  • We can determine the side length of a cube from its volume by taking the cube root of the volume.

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