Estimate cube roots

Key Notes :

📌 Estimate Cube Roots

Understanding Cube Roots
  • The cube root of a number x is a number y such that:

y3=x

  • Symbolically, ∛x=y.
  • Example: ∛27=3 because 33=27.
Perfect Cubes
  • Perfect cubes are numbers like 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
  • Knowing perfect cubes helps in estimating cube roots.
CubeCube Root
11
82
273
644
1255
2166
3437
5128
7299
100010
Steps to Estimate Cube Roots of Non-Perfect Cubes

1. Find two nearest perfect cubes:

  • Identify a perfect cube just smaller than the number.
  • Identify a perfect cube just larger than the number.

2. Use these cubes to estimate:

  • The cube root will lie between the roots of these two perfect cubes.

3. Refine estimate (optional):

  • Compare the number with cubes in-between to get a closer approximation.
Example 1 – Estimating ∛50
  1. Nearest perfect cubes: 27=33 and 64=43
  2. So, ∛50​ lies between 3 and 4.
  3. Check closer: 3.53=42.875, 3.63=46.656, 3.73=50.653
  4. ✅ Estimate: ∛50​≈3.7
Example 2 – Estimating ∛200
  1. Nearest perfect cubes: 125=53 and 216=63
  2. So, ∛200​ lies between 5 and 6.
  3. Check closer: 5.83=195.112, 5.93=205.379
  4. ✅ Estimate: ∛200 ≈ 5.85
Quick Tips
  • Memorize perfect cubes up to 103=1000 ✅
  • Use trial and error with decimals to refine your estimate
  • Cube roots of numbers less than 1 are between 0 and 1
  • Use a calculator for more accurate estimation, if needed
Practice Questions
  1. Estimate ∛90
  2. Estimate ∛15
  3. Estimate ∛300
  4. Estimate ∛0.125

Learn with an example

Which two integers is 3√49 between ?

  • 5 and 6
  • 15 and 16
  • 13 and 14
  • 3 and 4

Find the perfect cubes that are just below and just above 49.

The perfect cube just below 49 is 27.

3√27 = 3

The perfect cube just above 49 is 64 .

3√64 = 4

3√27 < 3√49 < 3√64 , so 3 <3√49 < 4.

Which two integers is 3√14 between ?

  • 12 and 13
  • 2 and 3
  • 6 and 7
  • 1 and 2

Find the perfect cubes that are just below and just above 14.

The perfect cube just below 14 is 8.

3√8 = 2

The perfect cube just above 14 is 27 .

3√27 = 3

3√8 < 3√14 < 3√27 , so 2 <3√14 < 3.

Which two integers is 3√122 between ?

  • 0 and 1
  • 4 and 5
  • 13 and 14
  • 11 and 12

Find the perfect cubes that are just below and just above 122.

The perfect cube just below 122 is 64.

3√64 = 4

The perfect cube just above 122 is 125 .

3√125 = 5

3√64 < 3√122 < 3√125 , so 4 <3√122 < 5.

Let’s practice!