Cube Root: Definition, Symbol, Methods to Calculate, and Solved Examples
A cube root of a number is the value that multiplies three times to equal the original. Learn methods to calculate the cube root of a number here.
Chandni 17 Jan, 2025
Share
Cube root: Suppose you are building a cube-shaped box and know its volume is 27 cubic units. How would you find the length of each side? The solution lies in the cube root .
A cube root works in reverse of cubing a number. For example, 3 × 3 × 3 = 27, so the cube root of 27 is 3. This simple yet powerful concept is used in real life to calculate dimensions, volumes, and much more. Let’s explore the concept of cube roots in detail here. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/Curious-Jr-Ad-3-1-1.mp4"][/video]What is a C ube?
Before discussing what cube roots are, let’s first understand what a cube is in math. A cube is the result of multiplying a number by itself three times. In other words, when a number is raised to the power of 3, it is said to be "cubed."Example:
The cube of 2 is calculated as: 2× 2 × 2 = 8 So, 8 is the cube of 2. Similarly, the cube of 4 is: 4× 4× 4=64 Hence, 64 is the cube of 4. In mathematical notation, the cube of a number n is written as n 3 . The small 3 indicates that the number is multiplied by itself three times. [video width="1920" height="1080" mp4="https://www.pw.live/exams/wp-content/uploads/2024/12/curious-jr.mp4"][/video]Unit Digits in Cube Numbers
The unit digit of a number's cube follows specific patterns based on whether the number is odd or even. Odd Numbers : If a number is odd, its cube will also have an odd unit digit. For example: The cube of 3 is 27, and the unit digit of 27 is 7 (odd). The cube of 5 is 125, and the unit digit of 125 is 5 (odd). Even Numbers : If a number is even, its cube will also have an even unit digit. For example: The cube of 2 is 8, and the unit digit of 8 is 8 (even). The cube of 6 is 216, and the unit digit of 216 is 6 (even). Now that we know what a cube is, let’s explore cube roots!Cube Root D efinition
A cube root of a number 'x' is a value 'y' such that when multiplied by itself three times (i.e., y × y × y), it equals x. Mathematically, this is expressed as: y 3 = x, where x is the cube of y. I n other words, the cube root of a number is the value that, when cubed, gives the original number For instance, the cube root of 512 is 8 because 8 × 8 × 8 = 512Cube Root Symbol
The cube root is represented by the symbol ‘ 3 √’ . Unlike the square root, which uses the symbol ‘√’ , the cube root includes a small 3 to indicate that it represents the root of a number that has been cubed.For example:
- The cube root of 5 is written as 3 √5
- The cube root of 11 is written as 3 √11
Cube Root Formula
The cube root formula helps to find the cube root of any given number. If the cube of a number m is n, it can be expressed as: m 3 = n To calculate the cube root of n, the formula is: n 1/3 = m This formula establishes the relationship between a number, its cube, and its cube root, making it useful in various mathematical calculations.| Number (a) | Cube (a³) | Cube Root (∛a) |
|---|---|---|
| 1 | 1 | 1.000 |
| 2 | 8 | 1.260 |
| 3 | 27 | 1.442 |
| 4 | 64 | 1.587 |
| 5 | 125 | 1.710 |
| 6 | 216 | 1.817 |
| 7 | 343 | 1.913 |
| 8 | 512 | 2.000 |
| 9 | 729 | 2.080 |
| 10 | 1000 | 2.154 |
| 11 | 1331 | 2.224 |
| 12 | 1728 | 2.289 |
| 13 | 2197 | 2.351 |
| 14 | 2744 | 2.410 |
| 15 | 3375 | 2.466 |
| 16 | 4096 | 2.520 |
| 17 | 4913 | 2.571 |
| 18 | 5832 | 2.621 |
| 19 | 6859 | 2.668 |
| 20 | 8000 | 2.714 |
How to Find the Cube Root of a Number?
Finding the cube root of a number can be done using two main methods:- Prime Factorization Method
- Estimation Method
Prime Factorization Method
This method involves breaking the number into its prime factors. A number is a perfect cube if all the prime factors can be grouped into sets of three. The cube root is found by taking one factor from each group.Steps to find cube root using prime factorization:
Step 1: Perform the prime factorization of the number. Step 2: Group the prime factors into sets of three. Step 3: Multiply one factor from each group to find the cube root. Example 1: Find the cube root of 729. Prime factorize 729: 729 = 3 × 3 × 3 × 3 × 3 ×3 Group the factors into sets of three: 729 = (3×3×3) × (3×3×3) Take one factor from each group and multiply: 3×3=9 The cube root of 729 is 9. Example 2: Find the cube root of 3375. Prime factorize 3375 : 3375=5 × 5 × 5 × 3 × 3 × 3 Group the factors into sets of three: 3375= (5×5×5) × (3×3×3) Take one factor from each group and multiply: 5×3=15 The cube root of 3375 is 15 .Estimation Method
The estimation method is used to find the cube root of larger numbers by grouping digits and using properties of cubes.Steps to find cube root using the estimation method
Step 1: Separate the number into groups of three digits, starting from the rightmost digit. Step 2: Use the last digit of the number to determine the unit digit of the cube root (based on known cube values). Step 3: Identify the range of cubes in which the remaining group of digits falls. Step 4: Combine the results to find the cube root. Tip: The following table can be helpful in determining the unit digit of a cube root| Last Digit of Number | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| Unit digit of Cube Root | 0 | 1 | 8 | 7 | 4 | 5 | 6 | 3 | 2 | 9 |
- From the cube table, the cube root of a number ending in 1 also ends in 1.
- The unit digit of the cube root is 1 .
| Related Articles | |
| Area of Rectangle | Isosceles Triangle |
| Composite Numbers | Differentiation |
| Perimeter of Rectangle | Surface Area of Cylinder |
</strong><b>Cube Root </b><strong>FAQs
What is a perfect cube?
A number is a perfect cube if its cube root is a whole number. For example, 27 is a perfect cube because the cube root of 27 is 3.
Can the cube root of a number be negative and positive?
For real numbers, the cube root is either negative or positive, depending on the sign of the original number.
What is the cube root of 0?
The cube root of 0 is 0, since 0 × 0 × 0=0
How is a cube root different from a square root?
A cube root is a number that, when cubed, gives the radical and can be negative for negative numbers. A square root is a number that, when squared, gives the radicand but cannot be negative for negative numbers in the real number system.
Talk to a counsellorHave doubts? Our support team will be happy to assist you!
Join 15 Million students on the app today!
Live & recorded classes available at easeDashboard for progress trackingMillions of practice questions at your fingertips
Free Learning Resources
PW Books
Notes (Class 10-12)
PW Study Materials
Notes (Class 6-9)
Ncert Solutions
Govt Exams
Class 6th to 12th Online Courses
Govt Job Exams Courses
UPSC Coaching
Defence Exam Coaching
Gate Exam Coaching
Other Exams
Know about Physics Wallah
Physics Wallah is an Indian edtech platform that provides accessible & comprehensive learning experiences to students from Class 6th to postgraduate level. We also provide extensive NCERT solutions, sample paper, NEET, JEE Mains, BITSAT previous year papers & more such resources to students. Physics Wallah also caters to over 3.5 million registered students and over 78 lakh+ Youtube subscribers with 4.8 rating on its app.
We Stand Out because
We provide students with intensive courses with India’s qualified & experienced faculties & mentors. PW strives to make the learning experience comprehensive and accessible for students of all sections of society. We believe in empowering every single student who couldn't dream of a good career in engineering and medical field earlier.
Our Key Focus Areas
Physics Wallah's main focus is to make the learning experience as economical as possible for all students. With our affordable courses like Lakshya, Udaan and Arjuna and many others, we have been able to provide a platform for lakhs of aspirants. From providing Chemistry, Maths, Physics formula to giving e-books of eminent authors like RD Sharma, RS Aggarwal and Lakhmir Singh, PW focuses on every single student's need for preparation.
What Makes Us Different
Physics Wallah strives to develop a comprehensive pedagogical structure for students, where they get a state-of-the-art learning experience with study material and resources. Apart from catering students preparing for JEE Mains and NEET, PW also provides study material for each state board like Uttar Pradesh, Bihar, and others
Copyright © 2025 Physicswallah Limited All rights reserved.
