How to Calculate Cube Root: Top 3 methods
- Author: Noreen Niazi
- Last Updated on: January 17, 2024
As a beginner in mathematics, it’s essential to master the basics, and one of the fundamental concepts you’ll come across is cube roots. Cube roots are essential to mathematics; mastering them is crucial for anyone looking to excel in math. This guide will look at cube roots, how to calculate them, and why they are important.
Introduction to Cube Roots
A cube root is a number that will give you the original number when multiplied by itself three times. In simpler terms, it is the inverse of cubing a number. For instance, the cube root of $$27$$ is $$3$$ since $$3$$ multiplied by itself three times $$(3 \times 3 \times 3)$$ gives you $$27$$. The symbol for cube root is ∛, and it is often used in mathematical equations and formulas.
What is a Cube Root, and How Does it Work? Introduction to Cube Roots
To understand cube roots better, let’s consider an example. Suppose you have a cube with a volume of $$125$$ cubic units.
To calculate the length of each of its sides, you need to find the cube root of $$125$$. In this case, the cube root of $$125$$ is $$5$$. This means that each side of the cube is $$5$$ units long.
The process of calculating the number that, when cubed, yields the original number is known as finding the cube root of a number. The opposite of cubing a number is what this technique entails. The cube root of an integer x, for instance, is a number y that is such that $$y^3 = x$$.
Why Do We Need to Calculate Cube Roots?
Cube roots are essential in various mathematical applications. For example, they are used in finding the volume of a cube, which is a fundamental concept in geometry. Additionally, cube roots are used in algebraic expressions and equations and are an essential part of calculus. Therefore, if you’re interested in pursuing a career in mathematics or any field that requires mathematical knowledge, it’s crucial to master cube roots.
Basic Steps for Calculating Cube Roots
To calculate the cube root of a number, you need to follow these basic steps:
- Determine the number whose cube root you want to find.
- Divide that number into groups of three digits, starting from the rightmost digit.
- Starting with the leftmost group of digits, find the largest cube that is less than or equal to that group of digits.
- Write down the cube root of that group of digits.
- Subtract the cube of the cube root from the group of digits and bring down the next group of digits.
- Repeat steps 3 to 5 until you have gone through all the groups of digits.
- The result is the cube root of the original number.
Examples of How to Calculate Cube Roots
Let’s look at examples of calculating cube roots using the steps outlined above.
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Example 1: How to calculate Cube Root of 216
Suppose we want to find the cube root of $$216$$. We’ll follow the steps outlined above:
- The number we want to find the cube root of is $$216$$.
- $$216$$ has only three digits, so we don’t need to divide it into groups.
- The largest cube less than or equal to $$216$$ is $$125$$, $$5^3$$.
- Therefore, the cube root of the first group of digits $$(216)$$ is $$5$$.
- We subtract $$125$$ from $$216$$, which gives us $$91$$. We bring down the next group of digits $$(00)$$.
- There are no more groups of digits to bring down, so we have completed the process.
- The cube root of $$216$$ is $$6$$.
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Example 1: How to calculate Cube Root of 64
Suppose we want to find the cube root of $$64$$. We’ll follow the same steps as in Example 1:
- The number we want to find the cube root of is $$64$$.
- $$64$$ has only two digits, so we don’t need to divide it into groups.
- The largest cube less than or equal to $$64$$ is $$27$$, $$3^3$$.
- Therefore, the cube root of the first group of digits $$(64)$$ is $$3$$.
- We subtract $$27$$ from $$64$$, which gives us $$37$$. We bring down the next group of digits $$(00)$$.
- There are no more groups of digits to bring down, so we have completed the process.
- The cube root of $$64$$ is $$4$$.
How to calculate cube root by prime factorization method.
The prime factorization method is one of the most common methods for calculating cube roots. If you dont know how calculate cube root with by prime factorization, then follow these simple steps.
First, break down the number you want to find the cube root into its prime factors and group them into three sets. The cube root is then found by taking one element from each group and multiplying them together.
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Example: 1 How to calculate cube root of 216.
Step 1: Write the prime factor of the cube root. To find the cube root of $$216$$, you would first break it down into its prime factors:
$$2 \times 2 \times 2 \times 3 \times 3 \times 3$$
You can also find prime factorization by the tree method.
Step 2: Then, you would group them into three sets: $$(2 \times 2 \times 2) $$ and $$(3 \times 3 \times 3)$$. Finally, you would take one factor from each group and multiply them together: $$2 \times 3 = 6$$ . Therefore, the cube root of $$216$$ is $$6$$ .
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Example: 2 How to calculate cube root of $$729$$.
Step 1: Write the prime factor of the cube root. To find the cube root of $$729$$, you would first break it down into its prime factors:
$$3\times 3 \times 3\times 3 \times 3 \times 3$$
You can also find prime factorization by the tree method.
Step 2: Then, you would group them into three sets: $$3\times 3\times 3$$ and $$3\times 3\times 3$$ . Finally, you would take one factor from each group and multiply them together: $$3\times 3=9$$ . Therefore, the cube root of $$729$$ is $$9$$.
How to calculate cube root by estimation
If you want to avoid using the prime factorization method or need a quick estimate, you can use estimation to find an approximate cube root.
Start by finding the perfect cubes closest to the number you want to see the cube root of.
For example, if you want to find the cube root of $$50$$, start by finding the perfect cubes closest to $$50$$, which are $$27$$ and $$64$$.
Then, estimate where $$50$$ falls between those two perfect cubes. In this case, it’s closer to $$27$$ than $$64$$. Finally, take the cube root of the perfect cube that it’s most relative to, which is $$27$$, and adjust your estimate accordingly.
In this case, the cube root of $$27$$ is $$3$$, so the approximate cube root of $$50$$ is slightly less than $$3$$.
Use a calculator or online tool for more complex cube roots.
Using a calculator or online tool may be helpful if you need to calculate a cube root that is more complex than simple whole numbers.
Many scientific calculators have a cube root function. You can find it by pressing the cube root button (usually labeled”∛”).
Many online calculators and tools can help you calculate cube roots, including Wolfram Alpha and Mathway. These tools can be beneficial if you compute cube roots with decimals or fractions.
However, it’s still important to understand the basic principles of cube roots to check your answers and ensure that the calculator or tool gives accurate results.
How to calculate Cube Roots of Negative Numbers
So far, we’ve only looked at cube roots of positive numbers. But what about negative numbers? Can we find the cube root of a negative number?
The answer is yes. We use the same formula but introduce a negative sign when dealing with negative numbers. For example, the cube root of $$-216$$ is $$-6$$ since $$-6 \times -6 \times -6 = -216$$.
How to calculate Cube Roots of Non-Perfect Cubes
We’ve also looked at examples of finding the cube root of perfect cubes, such as $$27$$ and $$64$$. But what about non-perfect cubes, such as $$20$$ or $$100$$?
Example 1: How to calculate Cube root of 20.
In this case, the cube root will not be a whole number, and you may need to use a calculator to find an approximate answer.
Example 2: How to calculate Cube root of 100.
In this case, the cube root will not be a whole number, and you may need to use a calculator to find an approximate answer.
Applications of Cube Roots in Math and Beyond
Cube roots have several applications in mathematics, including geometry, algebra, and calculus. Additionally, cube roots are used in fields such as engineering and physics, where they are used to calculate the volume of objects and to solve equations. Beyond math, cube roots are also used in computer science and cryptography, where they are used to encrypt and decrypt messages.
Practice and check your work to improve your skills.
Like any skill, calculating cube roots takes practice. The more you do it, the more comfortable and confident you will become.
You must check your work to ensure you are on the right track and catching any mistakes early on.
One way to check your work is to cube your answer and see if it matches the original number you were trying to find the cube root of.
Another way is to double-check your answer using a calculator or online tool.
With practice and careful checking, you’ll be able to calculate cube roots with ease.
Conclusion: Mastery of Cube Roots for Future Math Success
In conclusion, mastering cube roots is essential for anyone looking to excel in mathematics or any field that requires mathematical knowledge. By understanding what cube roots are, how to calculate cube root, and their applications in math and beyond, you’ll be well on your way to success in your studies and career. So, practice calculating cube roots, try using a cube root calculator, and work on mastering this fundamental concept.
FAQs
Q1: What is the root cube and why does it matter?
A1: Cube root is a mathematical term that denotes the number which, when multiplied by itself three times (cubed), leads to some specified value. It is instrumental in multiple disciplines ranging from geometry to engineering for solving issues involving volume and dimensions.
Q2: What is the standard procedure in calculating cubes?
A2: The general technique involves determining the number that, when multiplied by itself twice, results in the target value. This process may be done through the use of iterative procedures or by using a scientific calculate that can do cube root.
Q3: Are there any special shortcuts or mental tricks for finding cube roots?
A3: Mentally determining cube roots are not easy but some approval techniques and patterns can help in faster estimations. Nevertheless, rather accurate mental calculations often require a good grasp of number characteristics.
Q4: Can you guide me through how to calculate cube roots manually?
A4: Certainly! Estimation, iterative methods and sometimes logarithms are usually involved on step-by-step instructions. These methods are variable depending on the level of accuracy that is needed.
Q5: What is the difference between cube root and square
A5: The cube root is a value that, when multiplied by itself two times, gives you the given quantity (x^(1/3)). On the other hand, a square root is that number which, when squared gives you x.
Reference: How to calculate Cube Root
- Math Is Fun. “Cube Root.” (https://www.mathsisfun.com/numbers/cube-root.html)
- Khan Academy. “Cube Roots.” (https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:rational-exponents-radicals/x2f8bb11595b61c86:radicals-intro/v/roots-introduction)
- Purplemath. “Cube Roots.” (https://www.purplemath.com/modules/roots.htm)
- Calculator Soup. “Cube Root Calculator.” (https://www.calculatorsoup.com/calculators/algebra/cube-root-calculator.php)
- Brilliant. “Cube Root of a Perfect Cube.” (https://brilliant.org/wiki/cube-root/)