Finding the cube root of a large number is difficult. Simple shortcuts make the procedure faster and easier. These shortcuts use basic patterns and straightforward steps to find the cube root without complex calculations or advanced tools.
This method is useful for saving time and improving mental math skills. By learning these techniques, one can easily determine the cube root of any number.
What is Cube Root?
Cube root is a number that when multiplied three times (or cubed) by itself, equals the given number. For Example: 3 multiplied by itself three times i.e. (3 x 3 x 3) = 27, therefore, the cube root of 27 is 3.
In mathematical terms, if x is the cube root of y, then x3 = y. The cube root is denoted using the radical symbol with a small 3 above it, i,e,: ∛y.
Table of Cube Root Numbers
Cube roots of some common numbers are given below:
Number
| Cube root
|
|---|
|
1
|
1
|
|
8
|
2
|
|
27
|
3
|
|
64
|
4
|
|
125
|
5
|
|
216
|
6
|
|
343
|
7
|
|
512
|
8
|
|
729
|
9
|
|
1000
|
10
|
Short Tricks to Find Cube Root
Any integer can have its cube root difficult to find, however there are various methods and shortcuts to make the job easier. Here are a few methods:
Digit by Digit Method
- Step 1: Find the digital root of the number by repeatedly summing the digits until you get a single digit. For example, the digital root of 1728 (1 + 7 + 2 + 8 = 18, 1 + 8 = 9).
- Step 2: Use the digital root to narrow down the possibilities, as the cube root of a number often has a specific digital root pattern. While this method is not always precise, but it can help in estimating.
Using Prime Factorization
- Step 1: Factor the number into its prime factors. For example, for 216, the factorization is 23×33.
- Step 2: Group the factors into triples. Here, 23 and 33are already grouped.
- Step 3: Take one factor from each group. The cube root of 23×33 is 2×3=6.
Estimation Method
- Step 1: Find two perfect cubes between which the number lies. For example, if you want to find the cube root of 50, you know it lies between 33=27 and 43=64.
- Step 2: Estimate the value between these two numbers. Since 50 is closer to 64 than to 27, you can guess the cube root to be closer to 4.
- Step 3: To obtain a closer estimate, average your prediction or use trial and error.
Shortcut Method to Find Cube Roots of Perfect Cubes
To find the cube root of perfect cubes by the shortcut method we should first know the unit value of the cube root of a number. The table given below provides the list of the corresponding unit digit according to the unit digit of the given number.\
Unit Digit of the number
| Unit digit of its cube root
|
|---|
|
0
|
0
|
|---|
|
1
|
1
|
|---|
|
2
|
8
|
|---|
|
3
|
7
|
|---|
|
4
|
4
|
|---|
|
5
|
5
|
|---|
|
6
|
6
|
|---|
|
7
|
3
|
|---|
|
8
|
2
|
|---|
|
9
|
9
|
|---|
From the above table we can note that the unit digit of the cube root of a number ending with 0, 1, 4, 5, 6, and 9 remains same whereas for a number ending with 2 will have a cube root with unit digit as 8 and vica-versa and for a number ending with 3 will have a cube root with unit place as 7 and vica-versa.
Let us now understand how to find the cube root of perfect cube numbers using a shortcut and a trick.
For example, we have a number 17576
Step 1: Divide the Number
- Separate the number into groups of three.
- For 17576, the last three digits are 576, and the remaining digits are 17.
Step 2: Determine the Unit Digit of the Cube Root
- Look at the last digit of the original number. For 17576, the last digit is 6.
- Numbers with a cube ending in 6 have a cube root that also ends in 6. Therefore, the unit digit of the cube root of 17576 is 6.
Step 3: Consider the Remaining Digits
- Consider the remaining number, in this case 17.
Get the biggest perfect cube that is either less than or equal to 17. Less than 17 perfect cubes are 1 and 8. 8 is the biggest here. - The cube root of 8 is 2.
Combine the Results
- Combine the findings of the earlier results. From the cube root of 17576, the unit digit is 6 and the tens digit is 2.
- 17576's cube root is therefore 26.
Read More,
Solved Examples
Example 1. Find the cube root of 12167.
Solution:
Step 1: Separate the number into two parts: the last three digits and the remaining digits.
- 12167: last three digits are 167, and remaining digits are 12.
Step 2: Determine the unit digit of the cube root:
- The last digit of 12167 is 7. Numbers ending in 7 have cube roots ending in 3
- So, the unit digit of the cube root is 3.
Step 3: Consider the remaining digits (12):
- Find the largest perfect cube less than or equal to 12 is 8.
- The cube root of 8 is 2.
The cube root of 12167 is 23.
Example 2: Estimate the cube root of 50,000.
Solution:
Step 1: Let us identify the nearest cubes to 50,000
- 33=27
43=64 - The cube root of 50,000 lies between 303 (27,000) and 403 (64,000).
Step 2: Estimate the value between these two numbers.
- 50,000 is closer to 64,000 than it is to 27,000, so the cube root will be closer to 40 than to 30.
Step 3: Refine the guess:
- Let us try 35:
353=42,875 (which is still less than 50,000) - Let us try 36:
363=46,656 (which is closer but still less than 50,000) - Let us try 37:
373=50,653 (which is slightly greater than 50,000) - So, by estimation, the cube root of 50,000 is approximately 37.
Practice Questions
Question 1: Estimate cube root of 20,000.
Question 2: Find the cube root of 91125 using the digit-by-digit method
Question 3: Estimate the cube root of 1,50,000
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