124 LearnersLast updated on May 26th, 2025
Cube Root of 686
A number we multiply by itself three times to get the original number is its cube root. It has various uses in real life, such as finding the volume of cube-shaped objects and designing structures. We will now find the cube root of 686 and explain the methods used.
What is the Cube Root of 686?
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, ∛686 is written as 686(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 686, then y3 can be 686. Since the cube root of 686 is not an exact value, we can write it as approximately 8.819.
Finding the Cube Root of 686
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 686. 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 number, we often follow Halley’s method. Since 686 is not a perfect cube, we use Halley’s method.
Cube Root of 686 by Halley’s Method
Let's find the cube root of 686 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 = 686;
x = 9
∛a ≅ 9((93 + 2 × 686) / (2 × 93 + 686))
∛686 ≅ 9((729 + 2 × 686) / (2 × 729 + 686))
∛686 ≅ 8.819
The cube root of 686 is approximately 8.819.
Common Mistakes and How to Avoid Them in the Cube Root of 686
Finding the perfect cube of a number without any errors can be a difficult task for the students. This happens for many reasons. Here are a few mistakes the students commonly make and the ways to avoid them:
Mistake 1
Trying to find perfect cube roots for non-perfect cube numbers.
Children always try to calculate an exact whole number for the cube root of numbers like 686, which are not perfect cubes.
For example, they assume that they would get an exact whole number for 686 like they get for 512 (since 512 is a perfect cube). To avoid this error, memorize that some numbers don't have a perfect cube root, that is, the cube root of 686 is approximately 8.819.
Cube Root of 686 Examples:
Problem 1
Imagine you have a cube-shaped toy that has a total volume of 686 cubic centimeters. Find the length of one side of the box equal to its cube root.
Side of the cube = ∛686 ≈ 8.82 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 8.82 units.
Problem 2
A company manufactures 686 cubic meters of material. Calculate the amount of material left after using 200 cubic meters.
The amount of material left is 486 cubic meters.
Explanation
To find the remaining material, we need to subtract the used material from the total amount: 686 - 200 = 486 cubic meters.
Problem 3
A bottle holds 686 cubic meters of volume. Another bottle holds a volume of 100 cubic meters. What would be the total volume if the bottles are combined?
The total volume of the combined bottles is 786 cubic meters.
Explanation
Let’s add the volume of both bottles: 686 + 100 = 786 cubic meters.
Problem 4
When the cube root of 686 is multiplied by 3, calculate the resultant value. How will this affect the cube of the new value?
3 × 8.82 ≈ 26.46 The cube of 26.46 ≈ 18,522.79
Explanation
When we multiply the cube root of 686 by 3, it results in a significant increase in the volume because the cube increases exponentially.
Problem 5
Find ∛(300 + 386).
∛(300 + 386) = ∛686 ≈ 8.82
Explanation
As shown in the question ∛(300 + 386), we can simplify that by adding them.
So, 300 + 386 = 686.
Then we use this step: ∛686 ≈ 8.82 to get the answer.
FAQs on 686 Cube Root
1.Can we find the Cube Root of 686?
2.Why is the Cube Root of 686 irrational?
3.Is it possible to get the cube root of 686 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 686
- 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, 2 × 2 × 2 = 8, therefore, 8 is a perfect cube.
- Exponent: The exponent form of the number denotes the number of times a number can be multiplied by itself. In 686(1/3), ⅓ is the exponent which denotes the cube root of 686.
- Radical sign: The symbol 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 686 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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
