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101 LearnersLast updated on December 15, 2025
Cube Root of 67
A number we multiply by itself three times to get the original number is its cube root. It has various applications in real life, such as determining the volume of cube-shaped objects and designing structures. We will now find the cube root of 67 and explain the methods used.
What is the Cube Root of 67?
We have learned the definition of the cube root.
Now, let’s learn how it is represented using a symbol and exponent.
The symbol we use to express the cube root is the radical sign (∛), and the exponent we use is ⅓. In exponential form, ∛67 is written as 67^(1/3).
The cube root is just the opposite operation of finding the cube of a number.
For example: Assume ‘y’ as the cube root of 67, then y^3 can be 67. Since the cube root of 67 is not an exact value, we can write it as approximately 4.041.
Finding the Cube Root of 67
Finding the cube root of a number is to identify the number that must be multiplied three times resulting in the target number.
Now, we will go through the different ways to find the cube root of 67.
The common methods we follow to find the cube root are given below:
- Prime factorization method
- Approximation method
- Subtraction method
- Halley’s method
To find the cube root of a non-perfect cube number, we often follow Halley’s method. Since 67 is not a perfect cube, we use Halley’s method.
Cube Root of 67 by Halley’s method
Let's find the cube root of 67 using Halley’s method.
The formula is: ∛a ≅ x((x³ + 2a) / (2x³ + a)) where:
a = the number for which the cube root is being calculated
x = the nearest perfect cube
Substituting, a = 67;
x = 4
∛a ≅ 4((4³ + 2 × 67) / (2 × 4³ + 67)) ∛67 ≅ 4((64 + 134) / (128 + 67)) ∛67 ≅ 4.041
The cube root of 67 is approximately 4.041.
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Common Mistakes and How to Avoid Them in the Cube Root of 67
Finding the perfect cube of a number without any errors can be a difficult task for students.
This happens for many reasons.
Here are a few mistakes students commonly make and the ways to avoid them:
Mistake 1
Trying to find perfect cube roots for non-perfect cube numbers.
Children often try to calculate an exact whole number for the cube root of numbers like 67, which are not perfect cubes.
For example: They assume that they would get an exact whole number for 67 like they get for 64 (since ∛64 = 4). To avoid this error, memorize that some numbers don't have a perfect cube root. The cube root of 67 is approximately 4.041.
Mistake 2
Ignoring the exponent form
Most of us might forget that cube roots can also be written in exponent form.
For example: Forgetting that the cube root of 67 is 67^(1/3). To avoid this error, always learn the forms in which we can express the cube root of a number.
Mistake 3
Confusing cube root with division
Assuming the cube root of 67 is to divide 67 by 3.
For example: Mistakenly assume that 67/3 is the cube root of 67. To avoid this, children should remember that the cube root of 67 is the number they multiply by itself three times to get 67.
Mistake 4
Using prime factorization instead of Halley’s method
Instead of using Halley’s method for non-perfect cubes, children end up using prime factorization.
For example, they might try to find the cube root of 67 by breaking it down into its prime factors. Remember that prime factorization can help analyze a number's factors, but it's not the best method for non-perfect cubes. Since 67 is a non-perfect cube, you can use either the approximation method or Halley’s method.
Mistake 5
Rounding too early
Rounding the cube root too early in the process can lead to errors in subsequent steps.
For example: we might round the cube root of 67 to 4.0 too early in the calculation. To avoid this, ensure the whole calculation is done, then round the final result.
Cube Root of 67 Examples:
Problem 1
Imagine you have a cube-shaped toy that has a total volume of 67 cubic centimeters. Find the length of one side of the toy equal to its cube root.
Side of the cube = ∛67 ≈ 4.041 units
Explanation
To find the side of the cube, we need to find the cube root of the given volume.
Therefore, the side length of the cube is approximately 4.041 units.
Problem 2
A company manufactures 67 cubic meters of material. Calculate the amount of material left after using 15 cubic meters.
The amount of material left is 52 cubic meters.
Explanation
To find the remaining material, we need to subtract the used material from the total amount: 67 - 15 = 52 cubic meters.
Problem 3
A bottle holds 67 cubic meters of volume. Another bottle holds a volume of 10 cubic meters. What would be the total volume if the bottles are combined?
The total volume of the combined bottles is 77 cubic meters.
Explanation
Explanation: Let’s add the volume of both bottles: 67 + 10 = 77 cubic meters.
Let's say a substance in a chemical reaction has a concentration of 67 grams per cubic meter.
Calculate the new concentration if 8 grams per cubic meter are added to it.
The new concentration is 75 grams per cubic meter.
To find the new concentration, add the increase in concentration to the original value: 67 + 8 = 75 grams per cubic meter.
Problem 4
When the cube root of 67 is multiplied by 3, calculate the resultant value. How will this affect the cube of the new value?
3 × 4.041 = 12.123 The cube of 12.123 ≈ 1,782.62
Explanation
When we multiply the cube root of 67 by 3, it results in a significant increase in volume because the cube increases exponentially.
Problem 5
Find ∛(33 + 34).
∛(33 + 34) = ∛67 ≈ 4.041
Explanation
As shown in the question ∛(33 + 34), we can simplify that by adding them.
So, 33 + 34 = 67.
Then use this step: ∛67 ≈ 4.041 to get the answer.
FAQs on 67 Cube Root
1.Can we find the Cube Root of 67?
2.Why is the Cube Root of 67 irrational?
3.Is it possible to get the cube root of 67 as an exact number?
4.Can we find the cube root of any number using prime factorization?
5.Is there any formula to find the cube root of a number?
Important Glossaries for Cube Root of 67
- Cube root: The number that is multiplied three times by itself to get the given number is the cube root of that number.
- Perfect cube: A number is a perfect cube when it is the product of multiplying a number three times by itself. A perfect cube always results in a whole number. For example: 3 × 3 × 3 = 27, therefore, 27 is a perfect cube.
- Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In 67(1/3), ⅓ is the exponent which denotes the cube root of 67.
- Radical sign: The symbol that is used to represent a root is expressed as (∛).
- Irrational number: Numbers that cannot be put in fractional forms are irrational. For example, the cube root of 67 is irrational because its decimal form goes on continuously without repeating the numbers.
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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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: He loves to play the quiz with kids through algebra to make kids love it.
