Cube Root by Prime Factorization Method (With Solved Example)
The concept of how to find cube root plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding cube roots helps students tackle numbers smartly, ace competitive exams, and solve practical problems about area, volume, and measurement.
What Is How to Find Cube Root?
A cube root is defined as the number that, when multiplied by itself twice (three times in total), gives back the original number. For example, the cube root of 27 is 3, since 3 × 3 × 3 = 27. You’ll find this concept applied in areas such as algebra, geometry, and real-world measurement problems.
Key Formula for How to Find Cube Root
Here’s the standard formula: \( \sqrt[3]{x} = y \) where \( y \times y \times y = x \). The cube root symbol is ∛, and it can also be written as \( x^{1/3} \).
Cross-Disciplinary Usage
How to find cube root is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE, NTSE, or Olympiads often encounter cube root calculations in different subjects or real world contexts.
Methods to Find Cube Root
There are several ways to find the cube root of a number, whether it's a perfect cube, non-perfect cube, or even a decimal. The main methods are:
- Prime Factorization Method
- Estimation (Short-Trick) Method
- Division Method (for large/non-perfect cubes or decimals)
- Calculator or cube root tables (for quick lookup or irrational numbers)
Step-by-Step Illustration: How to Find Cube Root by Prime Factorization
- Write the number as a product of its prime factors.
- Group the prime factors in sets of three identical numbers.
- Take one number out of each group.
- Multiply the selected numbers.
Example: Find the cube root of 216.
2. Group in triples: (2 × 2 × 2) and (3 × 3 × 3)
3. One from each: 2 and 3
4. Multiply: 2 × 3 = 6
Final Answer: The cube root of 216 is 6.
Step-by-Step Illustration: How to Find Cube Root by Estimation Method
- Divide the number into groups of three digits from right (for large numbers or non-perfect cubes).
- Find the cube root of the group on the left using a mental cube root table (see below).
- Estimate the unit’s digit of the root from the group on the right (using cube endings pattern).
- Combine the estimates for the answer.
Example: Find the cube root of 74088.
2. Estimate first digit (from 074): Cube root of 64 is 4 and of 125 is 5. 74 is closer to 64, so tens digit is 4.
3. Last digit (from 088): 8 at the end means root ends in 2 (since ∛8 = 2).
4. Combine: 42.
Final Answer: The cube root of 74088 is 42.
How to Find Cube Root of a Decimal or Negative Number
For decimal numbers, move the decimal point to make the number a whole number (multiply/divide by 1000s), find the root, then put the decimal as required. For negative numbers, the cube root will also be negative (for example, ∛−27 = −3).
Cube Root Table (1–20)
| Number | Cube | Cube Root |
|---|---|---|
| 1 | 1 | 1 |
| 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 |
| 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 |
Speed Trick or Cube Root Shortcut
Here’s a quick cube root trick for perfect cubes:
- Look at the last digit of the number. Use this pattern:
Last digit of number Unit digit of Cube Root 1 1 8 2 7 3 4 4 5 5 6 6 9 9 0 0 - Divide other digits (except last 3) and estimate which integer’s cube is just less.
Tricks like this will save you time in Maths exams. You can find more such shortcuts on Vedantu’s Cube Root Table and in Vedantu live maths sessions.
Try These Yourself
- Find the cube root of 1331 using the factorization method.
- Estimate the cube root of 1000 without a calculator.
- Find the cube root of 0.008.
- Identify if 32 is a perfect cube.
Frequent Errors and Misunderstandings
- Mixing up cube root and square root operations.
- Forgetting to group factors in sets of three in the factorization method.
- Not placing the decimal correctly when working with decimal numbers.
Relation to Other Concepts
The idea of how to find cube root connects closely with square roots, exponent rules, and cube numbers. Mastering this helps with understanding volume, powers, and roots in advanced maths topics. You may also refer to Square Root Finder and Cube Root of Numbers for more practice.
Classroom Tip
A quick way to remember how to find cube root is to "think in sets of three!" Always group three of the same factors for perfect cubes. Vedantu’s teachers often use visuals for this in class and encourage memorizing cubes from 1 to 20 for instant problem solving.
We explored how to find cube root—from definition, formula, methods, tricks, and its relation to other maths concepts. Practice more with Vedantu and use the live cube root table or cube root factorization guide for deeper mastery. Keep learning and you’ll become confident at solving cube root questions in any exam!
FAQs on How to Find Cube Root Step by Step
1. What is a cube root?
The cube root of a number is a value that, when multiplied by itself three times, equals the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. It's represented by the symbol ∛ or as a number raised to the power of ⅓ (e.g., 27⅓ = 3).
2. How do I find the cube root of a perfect cube using prime factorization?
To find the cube root using prime factorization:
• **Factorize** the number into its prime factors.
• **Group** the identical prime factors into sets of three.
• **Multiply** one factor from each group to obtain the cube root. For example, to find the cube root of 216: 216 = 2 × 2 × 2 × 3 × 3 × 3 = (2 × 3)3 = 63. Therefore, the cube root of 216 is 6.
3. How can I estimate the cube root of a number?
Estimating cube roots involves identifying the perfect cubes closest to your target number. For instance, to estimate the cube root of 100, note that 43 = 64 and 53 = 125. Since 100 is closer to 125, the cube root is approximately 5.
4. How do I calculate the cube root using the long division method?
The long division method for cube roots is more complex and involves a series of iterative steps. It's best explained with visual examples and step-by-step instructions. Refer to detailed resources online for a comprehensive guide to this method. It's less commonly used than estimation or prime factorization for simple problems.
5. What is the cube root of a negative number?
The cube root of a negative number is negative. For example, the cube root of -8 is -2 because (-2) × (-2) × (-2) = -8.
6. How do I find the cube root of a decimal number?
Finding the cube root of a decimal number often involves approximation methods or the use of calculators. One approach is to convert the decimal to a fraction and then use prime factorization if possible. For more precise results, utilize a calculator.
7. Are there any shortcuts or tricks for finding cube roots?
Memorizing the cubes of numbers from 1 to 10 can speed up calculations. Observing patterns in the last digits of cubes can also help estimate the cube root’s last digit. However, for complex numbers, calculators are highly recommended.
8. How is the cube root used in real-world applications?
Cube roots find applications in various fields, including determining the volume of a cube given its side length, calculating the side length of a cube from its volume, and in certain formulas in physics and engineering.
9. What's the difference between a cube root and a square root?
A **square root** finds a number that, when multiplied by itself once, equals the original number (e.g., √9 = 3 because 3 × 3 = 9). A **cube root** finds a number that, when multiplied by itself twice, equals the original number (e.g., ∛27 = 3 because 3 × 3 × 3 = 27).
10. How do I solve cube root word problems?
Solving cube root word problems involves translating the problem's description into a mathematical equation involving cube roots. Identify the unknown quantity, set up the equation, and then solve for the unknown using appropriate methods.
11. What are some common mistakes students make when finding cube roots?
Common errors include incorrect prime factorization, misinterpreting the grouping of factors in the prime factorization method, and inaccurate estimations when using approximation techniques. Careless calculations and misunderstanding of negative cube roots are also frequent issues.
12. Can I use a calculator to find cube roots?
Yes, most scientific and even basic calculators have a cube root function (often denoted as ∛ or x1/3). Simply input the number and press the appropriate button for a direct result. This is the most efficient method for non-perfect cubes and decimals.
