One of the most interesting topics in mathematics is the cube root of a number. Understanding cube roots helps students improve their logical thinking and number sense. Cube roots also help build a strong foundation for solving high-level math and real-life problems involving volume, geometry, and algebra.
The cube root is the reverse process of finding a cube – the number multiplied by itself three times. For example, the cube root of 8 is 2 because 2 x 2 x 2 = 8.
However, when students reach higher grades or start preparing for competitive exams, they are stuck with the question—how to find the cube root of a number quickly or how to calculate the cube root of a number without a calculator? That’s where Vedic Maths comes in to help with cube root tricks and shortcuts.
In this blog, we will explore the cube root — how it works, methods for quickly finding it with step-by-step calculations, tips, and shortcuts.
Key Takeaways
- Cube roots are the inverse of cubes — they help to find the number that, when multiplied by itself three times, gives the original value.
- Understanding cube roots improves logical thinking and number sense, helping students solve higher-level problems in algebra, geometry, and physics.
- Geometrically, the cube root represents the side of a cube whose volume is known — useful in measurement, design, and engineering.
- The Vedic Maths cube root tricks (Vilokanam Sutra – “By Observation”) help to find cube roots quickly using simple last-digit patterns.
- Cube Root Pattern Tip:
- If the cube ends in 1, 4, 5, 6, 9, or 0 → the cube root ends with the same digit.
- If a cube ends in 2 → root ends with 8, and if it ends in 3 → root ends with 7 (and vice versa).
- Memorizing cubes from 1³ to 20³ strengthens number memory and speeds up mental math.
- Cube roots have real-life applications in geometry, architecture, physics, coding, and everyday problem-solving involving volume and 3D models.
- Master cube root tricks through The Vedic Maths Live Classes or Vedic Maths Bundle Kit by Mamta Rupesh, with free downloadable cube root worksheets PDFs.
1. Introduction: Why Learn Cube Roots the Smart Way
The cube root is one of the most important mathematical concepts taught from grades 6 to 12, and forms the foundation for topics such as algebra, geometry, and high-level mathematics.
Whether you are a student preparing for school exams or a competitive exam aspirant, mastering the cube root number is important to solve problems involving volume, powers, and equations.
However, many students find cube roots confusing and time-consuming, especially when dealing with large numbers and decimals. That's where Vedic Maths plays a prominent role.
Using Vedic Maths cube root tricks, you will learn how to calculate the cube root of a number quickly and accurately without a calculator.
2. What Is a Cube Root? (Simple Definition & Concept)
The cube root of a number is the reverse process of finding its cube. In simple terms, if a number multiplied by itself three times yields another number, then that first number is called the cube root of the resulting number.
For example:
2×2×2=8
So, the cube root of 8 is 2.
In mathematical form, ∛8 = 2
If you cube the result (i.e., multiply it by itself three times), you get the original number.
Remember, cube roots and cubes are inverses of each other.
Geometric Understanding
In geometry, a cube root helps you find the side length of a cube when its volume is known.
For example, if the volume of a cube is 125 cm³, then its side is ∛125 = 5 cm.
This visualization helps students relate cube roots to real-world 3D objects such as dice, Rubik’s cubes, boxes, and building blocks.
Why cube roots matter in math?
Understanding cube roots strengthens a student’s spatial reasoning and logical thinking. It lays a strong foundation for advanced math topics such as algebra, geometry, and physics, where cube roots are widely used in formulas for volume, density, and equations.
Remember, the cube root always tells you the length of one side of the cube.
Check out the NCERT Class 6 - Cube and Cube Roots chapter.
Check out the Vedic Maths Cube and Cube Roots course here.
3. How to Calculate Cube Root of a Number – Step-by-Step
Before learning shortcuts, let's learn about the standard cube root method to build conceptual clarity and how cubing the cube root resets the number when multiplied by itself three times to give the original value.
Let's learn how to find the cube root number using the long method.
Step-by-step method to find the cube root (long method)
Example 1: Find the cube root of 512
Step 1: Group digits into sets of three.
Start grouping digits from the right side of the number — each group contains three digits.
For 512, group as (512).
For larger numbers, say 12,167,000, group as (12)(167)(000).
Step 2: Find the largest cube smaller than the first group
Find the cube of a number less than or equal to the first group.
- The cube of 1 = 1
- The cube of 2 = 8
- The cube of 3 = 27
- The cube of 4 = 64
- The cube of 5 = 125
- The cube of 6 = 216
- The cube of 7 = 343
- The cube of 8 = 512
Here, the nearest cube ≤ 512 is 512 itself, which is 8³.
Step 3: Write the cube root
Since 8³ = 512,
∛512 = 8, which means the cube root of 512 is 8.
Example 2: Find the Cube Root of 2197
Step 1: Group digits → (2)(197)
Step 2: Find the largest cube smaller than the first group
- The cube of 1 = 1
- The cube of 2 = 8
- The cube of 3 = 27
- The cube of 4 = 64
- The cube of 5 = 125
- The cube of 6 = 216
- The cube of 7 = 343
- The cube of 8 = 512
- The cube of 9 = 729
- The cube of 10 = 1000
- The cube of 11 = 1331
- The cube of 12 = 1728
- The cube of 13 = 2197
Step 3: Write the cube root
Since 13³ = 2197,
∛2197 = 13, which means the cube root of 2197 is 13.
This traditional cube root method helps students understand how cubes and roots relate numerically — an important skill for Math Olympiad, NTSE, and CBSE/ICSE Board exams.
Read our blog, "Vedic Maths for Multiplication," to learn how to perform quick multiplication.
Once this concept is clear, students can move on to Vedic Maths cube root shortcut method to find cube roots quickly and mentally, without writing multiple steps.
4. Cube Root Shortcut Tricks Using Vedic Maths
One of the fastest and easiest ways to find cube roots, especially perfect cubes, is through Vedic Maths cube root tricks using the Vilokanam Sutra, meaning "By Observation".
Know more about Vedic Sutras here.
This powerful sutra helps you to calculate cube roots mentally within seconds, without a calculator.
The concept of Vilokanam Sutra — By Observation
When you look at a perfect cube (like 64 or 512), notice how its last digit always gives a hint about the cube root.
Vedic Maths helps you to recognize this pattern to find cube roots instantly — especially for numbers under 1000 (i.e., cubes from 1³ to 9³).
For example:
- The Cube of 4 = 64 → Cube root of number 64 = 4
- The Cube of 8 = 512 → Cube root of number 512 = 8
This observation-based trick is simple yet powerful once you master recognizing the patterns.
Pattern Recognition: The Last Digit Cube Root Table:
| Last Digit of the Cube | Cube Roots End With |
|---|---|
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |
| 3 | 7 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 3 |
| 8 | 2 |
| 9 | 9 |
Let's recognize the patterns from this table:
- The digits 1, 4, 5, 6, 9, 0 have the same last digit in both cube and cube root.
- The digits 2 ↔ 8 and 3 ↔ 7 are reversed pairs — meaning, if the cube ends in 2, its cube root ends in 8 (and vice versa).
This simple pattern helps you find the unit digit of the cube root instantly.
For example,
Let's find the cube root of a number 512 using the Vedic Maths technique/sutra
Step 1: Look at the last digit:
512 ends in 2, so the cube root must end in 8 (as per the pattern table).
Step 2: Ignore the last three digits and look at the remaining number.
Before 512, there's nothing, so consider it as 0.
Step 3: The cube of 8³ = 512, so the cube root is 8.
Answer: ∛512 = 8.
Another example,
Find ∛2744
Step 1: The last digit is 4, so the cube root ends in 4.
Step 2: Remove the last three digits → the remaining part is 2.
Step 3: Find the largest cube less than or equal to 2 → that's 1³ = 1, so the cube root's tens digit is 1.
Step 4: Combine → cube root is 14.
Answer: ∛2744 = 14.
To calculate cube roots, Vedic Math helps with pattern recognition (use this table) and place value estimation (remaining digits), which improves problem-solving ability.
5. Examples: Cube Root of 4, 8, 27, and More
Here are a few step-by-step solved examples of cube roots useful for students (grades 6 - 12 ) to learn cube root tricks.
What does cube root mean?
The cube root of N is a number x such that x³ = N . Math Notation: ∛N.
Examples:
1. ∛8
- Recognize: 2 x 2 x 2 = 8
- So
- ∛8 = 2.
- Check 2³ = 8.
2. ∛27
- Recognize: 3 x 3 x 3 = 27
- So ∛27 = 3
- Check 3³ = 27.
3. ∛125
- Recognize: 5 x 5 x 5 = 125
- So ∛125 =5
- Check 5³ = 125.
A few more quick cube roots:
∛1 = 1
∛2 = 1.259921
∛64 = 4 , as 4³ = 64
∛1000 = 10, as 10³ = 1000
Tips and Vedic Maths shortcuts
- Spot nearby shortcuts: If N is close to a perfect cube a³, start with a and adjust.
- Factor Approach: If N = a³ x k, then ∛N = a x ∛k — helpful when N has exact cubic factors.
- Use Binomial expansion for minor adjustments: (a + b)³= a³ + 3a²b + 3ab² + b³. This helps in expanding cube roots mentally for numbers near a known cube.
- Memorize cubes: Memorize 1³ to 20³ — very useful for quick cube calculation and initial guess.
6. Cube Root Table (1 to 20) for Quick Reference
When you are learning cube roots, referring to a quick cube root table helps you to memorize and estimate values faster, especially when solving math problems.
This table covers perfect cubes from 1³ to 20³, making it easier for students to visualize and understand the relation between numbers and their cube roots.
| Number (N) | Cube Root (∛N) |
|---|---|
| 1 | 1 |
| 8 | 2 |
| 27 | 3 |
| 64 | 4 |
| 125 | 5 |
| 216 | 6 |
| 343 | 7 |
| 512 | 8 |
| 729 | 9 |
| 1000 | 10 |
| 1331 | 11 |
| 1728 | 12 |
| 2197 | 13 |
| 2744 | 14 |
| 3375 | 15 |
| 4096 | 16 |
| 4913 | 17 |
| 5832 | 18 |
| 6859 | 19 |
| 8000 | 20 |
Importance of the Cube Root Table
- Strengthen number sense and help in quick estimation.
- Helpful for school exams, math competitive exams, and Olympiads.
- Help in mastering the cube root of numbers up to 20.
Read our blog, "Understanding the Cube of a Number,"to know about cubes and to refer to the 1 to 20 cubes table.
Download free cube root worksheets in PDF format today, designed by experienced math educator Mamta Rupesh. These worksheets contain practice exercises in various question formats, such as multiple-choice, fill-in-the-blanks, and real-world problems, and include key answers at the end to evaluate independently, identify mistakes, and correct them at the earliest.
7. Real-Life Applications of Cube Roots
Understanding cube roots isn't just about solving math problems; it is about learning how to connect math to real life. The cube root number helps us calculate volume, measure side or base lengths, find density, etc. The concept is widely used in the field of science, architecture, and coding.
Let us see a few real-life applications of cube roots
1. Geometry and Measurement
In geometry, cube roots are used to calculate the side length of a cube. For example, if the volume of a cube is 729 cm³, then its side is ∛729 = 9 cm.
This helps students understand spatial reasoning and 3D shapes — topics covered in the NCERT Class 8 and 9 math curriculum.
2. Architecture and Design
Architects use cube root calculation to maintain scale and proportion when designing 3D spaces, models, or buildings.
For example, when designing a room with a specific volume, finding the cube root helps estimate the optimal side length for a balanced design.
3. Physics and Engineering
In physics, cube roots are used in formulas involving density (mass/volume) and in calculations of pressure and volume for gases and solids.
For example, if you know the density and mass of a cube, you can use the cube root to find its side length.
4. Coding and Algorithms
Programmers often use cube roots in 3D simulations, graphics, and data modeling to create realistic structures.
In computer graphics, finding the cube root helps efficiently define coordinate and voxel (3D pixel) sizes.
5. Everyday Problem Solving
From calculating the edge of a water tank to scaling up 3D-printed designs, cube roots help connect classroom learning to daily reasoning and logical thinking.
8. Learn Cube Root Tricks with The Vedic Maths Courses
Learn how to find the cube root of a number quickly in just seconds using Vedic Maths techniques. Learn Vedic Maths in just 40 days using our Live Vedic Maths Classes. It is taught by Ms. Mamta Rupesh, a certified Vedic Maths teacher with over 8 years of experience, who has trained 15,000 students, 5,000 parents, and 5,000 teachers, and continues to train across the globe.
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9. Conclusion
Mastering cube roots doesn't have to be confusing or time-consuming when you understand the concept and learn Vedic Maths cube root tricks.
From finding cube roots of simple numbers such as 8 or 27 to calculating larger numbers such as 2197 or 6859, Vedic Maths makes it easier, faster, and accurate with its techniques. It is all about understanding the pattern and applying the correct method.
Whether you are a student preparing for school exams, Olympiads, or a competitive exam aspirant, or a parent helping a child in clearing cube roots, or a teacher looking to teach complex calculations more simply, these cube root techniques help to simplify number evaluation.
So, learn Vedic Maths techniques through our live classes or with the Vedic Maths bundle kit today, along with our free printable cube root worksheets PDFs, and speed up your calculations.
Start your journey with Vedic Maths and learn how to find the cube root of a number quickly within seconds.
10. FAQs on Cube Root of a Number
1What is the cube root of a number?
The cube root of a number is a value that, when multiplied by itself three times, gives the original number.
For example, since 2 × 2 × 2 = 8, the cube root of 8 is 2, written as ∛8 = 2.
2How are cube roots and cubes related?
Cube roots and cubes are inverse operations of each other.
If ∛x = y, then y³ = x.
Cube root formula ∛x = y.
In simple terms, one “undoes” the other — just like multiplication and division.
3What is the easiest way to find the cube root of a number?
The easiest way to find the cube root number is by using Vedic Maths cube root tricks, especially the Vilokanam Sutra (By Observation).
This method helps you calculate cube roots mentally by observing the last-digit pattern and using place-value estimation, making it ideal for quick calculations without a calculator.
4How to find the cube root of a number without a calculator?
You can use the long division method or Vedic Maths tricks.
In the long division method, group digits in threes (from right to left), find the nearest perfect cube, and calculate step by step.
In Vedic Maths, you can mentally identify the cube root using the last digit and cube pattern — perfect for exams and quick problem-solving.
5Why should students learn cube roots?
Learning cube roots improves logical reasoning, spatial visualization, and number sense.
It forms the foundation for geometry, algebra, physics, and engineering concepts like volume, density, and 3D modeling.
6What are some common cube roots students should memorize?
Here are a few you should always remember for quick calculations:
1³ = 1,
2³ = 8,
3³ = 27,
4³ = 64,
5³ = 125,
6³ = 216,
7³ = 343,
8³ = 512,
9³ = 729,
10³ = 1000.
7How is the cube root used in real life?
Cube roots are used to find the side length of a cube when the volume is known.
They are also used in architecture, engineering, coding, and 3D modeling to calculate dimensions, scale proportions, and structural designs efficiently.
8What is the difference between a cube and a cube root?
A cube is when a number is multiplied by itself three times (e.g., 3³ = 27).
A cube root is the reverse process — finding the number that, when cubed, gives the original number (∛27 = 3).
9How does Vedic Maths make cube roots easier?
Vedic Maths simplifies cube roots using logic-based patterns and observation rather than lengthy calculations.
It helps students find cube roots instantly, improve mental math skills, and build confidence in problem-solving.
You can join The Vedic Maths Live Classes or enroll in the Vedic Maths Bundle Kit by Ms. Mamta Rupesh, a certified educator with over 8 years of experience.
Visit www.thevedicmaths.com or call +91-866-056-2614 to book your free demo class today.
10What are the cube root shortcut patterns in Vedic Maths?
- If a cube ends in 1, 4, 5, 6, 9, or 0, the cube root ends with the same digit.
- If a cube ends in 2, the cube root ends with 8.
- If it ends in 3, the cube root ends with 7 (and vice versa).
- This simple trick helps find cube roots mentally within seconds.
