Cube Root by Estimation
While prime factorisation gives the exact cube root of a perfect cube, many numbers are not perfect cubes. For such numbers, you can estimate the cube root by identifying which two consecutive perfect cubes the number lies between.
For large perfect cubes, the estimation method provides a quick way to find the cube root without full factorisation — by analysing the number of digits and the unit digit of the number.
This method relies on your knowledge of cubes from 1³ to 10³ (or beyond) and the unit-digit patterns of cubes.
What is Cube Root by Estimation?
Definition: Estimation of cube root is a method to find the approximate or exact cube root of a number by narrowing it down between two known perfect cubes.
If a³ < N < b³, then a < ∛N < b
Perfect cubes to memorise:
- 1³ = 1, 2³ = 8, 3³ = 27, 4³ = 64, 5³ = 125
- 6³ = 216, 7³ = 343, 8³ = 512, 9³ = 729, 10³ = 1000
Methods
Method 1: For non-perfect cubes (estimation between consecutive cubes)
- Identify the two consecutive perfect cubes that the number lies between.
- The cube root lies between the corresponding integers.
- For a closer estimate, check how close the number is to each cube.
Example: Estimate ∛50.
- 3³ = 27 and 4³ = 64
- 27 < 50 < 64, so 3 < ∛50 < 4
- 50 is closer to 64 than 27, so ∛50 ≈ 3.7
Method 2: For large perfect cubes (NCERT method)
- Start from the units digit of the number. Using the unit-digit pattern of cubes, determine the units digit of the cube root.
- Strike off the last three digits of the number. Look at the remaining number.
- Find which two consecutive cubes this remaining number lies between. The smaller one gives the tens digit of the cube root.
Example: Find ∛17576.
- Step 1: Units digit of 17576 is 6. Since 6³ ends in 6, the units digit of ∛17576 is 6.
- Step 2: Strike off last 3 digits: 17576 → remaining = 17.
- Step 3: 2³ = 8 ≤ 17 < 27 = 3³. So tens digit = 2.
- ∛17576 = 26. Check: 26³ = 17576 ✓
Solved Examples
Example 1: Example 1: Estimating a non-perfect cube
Problem: Estimate ∛100.
Solution:
- 4³ = 64 and 5³ = 125
- 64 < 100 < 125
- So 4 < ∛100 < 5
- 100 is closer to 125, so ∛100 ≈ 4.6
Answer: ∛100 ≈ 4.6 (actual: 4.6416...).
Example 2: Example 2: Estimation of ∛200
Problem: Estimate ∛200.
Solution:
- 5³ = 125 and 6³ = 216
- 125 < 200 < 216
- 200 is very close to 216, so ∛200 ≈ 5.8
Answer: ∛200 ≈ 5.8 (actual: 5.848...).
Example 3: Example 3: Large perfect cube — ∛2744
Problem: Find ∛2744 by estimation.
Solution:
- Step 1: Units digit of 2744 is 4. Cube root ends in 4.
- Step 2: Remove last 3 digits: 2744 → remaining = 2.
- Step 3: 1³ = 1 ≤ 2 < 8 = 2³. Tens digit = 1.
- Cube root = 14.
- Check: 14³ = 2744 ✓
Answer: ∛2744 = 14.
Example 4: Example 4: ∛9261
Problem: Find ∛9261 by estimation.
Solution:
- Step 1: Units digit = 1. Cube root ends in 1.
- Step 2: Remove last 3 digits: 9261 → remaining = 9.
- Step 3: 2³ = 8 ≤ 9 < 27 = 3³. Tens digit = 2.
- Cube root = 21.
- Check: 21³ = 9261 ✓
Answer: ∛9261 = 21.
Example 5: Example 5: ∛32768
Problem: Find ∛32768 by estimation.
Solution:
- Step 1: Units digit = 8. Cube root ends in 2 (since 2³ = 8).
- Step 2: Remove last 3 digits: 32768 → remaining = 32.
- Step 3: 3³ = 27 ≤ 32 < 64 = 4³. Tens digit = 3.
- Cube root = 32.
- Check: 32³ = 32768 ✓
Answer: ∛32768 = 32.
Example 6: Example 6: ∛74088
Problem: Find ∛74088 by estimation.
Solution:
- Step 1: Units digit = 8. Cube root ends in 2.
- Step 2: Remove last 3 digits: 74088 → remaining = 74.
- Step 3: 4³ = 64 ≤ 74 < 125 = 5³. Tens digit = 4.
- Cube root = 42.
- Check: 42³ = 74088 ✓
Answer: ∛74088 = 42.
Example 7: Example 7: ∛13824
Problem: Find ∛13824 by estimation.
Solution:
- Step 1: Units digit = 4. Cube root ends in 4.
- Step 2: Remove last 3 digits: 13824 → remaining = 13.
- Step 3: 2³ = 8 ≤ 13 < 27 = 3³. Tens digit = 2.
- Cube root = 24.
- Check: 24³ = 13824 ✓
Answer: ∛13824 = 24.
Example 8: Example 8: Estimating ∛15
Problem: Estimate ∛15 to one decimal place.
Solution:
- 2³ = 8, 3³ = 27
- 8 < 15 < 27, so 2 < ∛15 < 3
- 15 is roughly in the middle, slightly below. ∛15 ≈ 2.5
- Check: 2.5³ = 15.625 (close to 15)
Answer: ∛15 ≈ 2.5 (actual: 2.466...).
Example 9: Example 9: ∛175616
Problem: Find ∛175616 by estimation.
Solution:
- Step 1: Units digit = 6. Cube root ends in 6.
- Step 2: Remove last 3 digits: 175616 → remaining = 175.
- Step 3: 5³ = 125 ≤ 175 < 216 = 6³. Tens digit = 5.
- Cube root = 56.
- Check: 56³ = 175616 ✓
Answer: ∛175616 = 56.
Example 10: Example 10: ∛531441 (3-digit cube root)
Problem: Estimate ∛531441.
Solution:
- Units digit = 1. Cube root ends in 1.
- Remove last 3 digits: 531441 → remaining = 531.
- This is larger than 8³ = 512 and less than 9³ = 729.
- But this gives tens digit = 8, and remaining might have more digits.
- Actually for 6-digit numbers, cube root has 2 digits: remaining = 531.
- Wait — 6-digit number can have cube root up to 99 (99³ = 970299).
- remaining = 531. 8³ = 512 ≤ 531 < 729 = 9³. Tens digit = 8.
- Cube root = 81.
- Check: 81³ = 531441 ✓
Answer: ∛531441 = 81.
Real-World Applications
Real-world applications:
- Quick mental maths: Estimating cube roots without a calculator in exams.
- Engineering: Approximate volume calculations — if volume is known, estimate the side length.
- Science: Estimating dimensions when given cubic measurements.
- Competitive exams: Speed calculation of cube roots is tested in olympiads and aptitude tests.
Key Points to Remember
- For non-perfect cubes, estimate by finding the two consecutive cubes the number lies between.
- For large perfect cubes, use the NCERT estimation method: unit digit + remaining number.
- The unit digit of a cube root follows a fixed pattern based on the unit digit of the cube.
- Pairs (2,8) and (3,7) swap when cubing.
- Digits 0, 1, 4, 5, 6, 9 stay the same.
- This method works for cubes of 2-digit numbers (4 to 6-digit cubes).
- Always verify your answer by cubing the result.
- Memorise cubes of 1 to 10 for quick estimation.
Practice Problems
- Estimate ∛70 between two consecutive integers.
- Find ∛4913 by estimation.
- Find ∛19683 by estimation.
- Estimate ∛300 to one decimal place.
- Find ∛68921 by the estimation method.
- Find ∛103823 by estimation.
- Estimate ∛10 between two consecutive integers.
- Find ∛250047 by estimation.
Frequently Asked Questions
Q1. How do you estimate a cube root?
Find the two consecutive perfect cubes the number lies between. The cube root lies between those integers. For large perfect cubes, use the unit-digit method.
Q2. What is the NCERT estimation method?
Step 1: Find units digit of cube root from unit digit of the number. Step 2: Remove last 3 digits. Step 3: Find which cubes the remaining number lies between — that gives the tens digit.
Q3. Does this method work for all numbers?
The NCERT method works only for perfect cubes of 2-digit numbers. For non-perfect cubes, you get an approximation between two integers.
Q4. Why do the digits 2 and 8 swap?
Because 2³ = 8 (ends in 8) and 8³ = 512 (ends in 2). So if a cube ends in 8, the cube root ends in 2, and vice versa.
Q5. How accurate is the estimation for non-perfect cubes?
You get the answer to the nearest integer. For more accuracy, you can interpolate or use the long division method.
Q6. What cubes should I memorise?
Memorise cubes of 1 to 15 at minimum: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375.
