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Estimation Method or Using Units Digit and Tens Digit

Recall that the units digit of the cube of a number depends upon the units digit of the given number
(see properties of cube numbers). The units digit of the cube root of a perfect cube number,
corresponding to the units digit of the number is given in the following table:

Units digit of the perfect cube number 0 1 2 3 4 5 6 7 8 9
Units digit of the cube root of the number 0 1 8 7 4 5 6 3 2 9

Note: We can find the cube roots of the perfect cubes having at most six digits by the method of estimation.
By observing the units digit of perfect cube numbers, we can determine the units digit of their
cube root. To find the cube root of a perfect cube, we follow the given steps:

Steps Example: Find the cube root of 551368 through
estimation.
1. Start making groups of three digits starting 551 368
from the right most digit of the number. Second group First group
2. In the first group, observe the digit at the units First group, i.e., in 368
place of the given perfect cube number and Units digit of the given number is 8
determine the digit at the units place of its cube ∴Units digit of the cube root of the given number = 2
root (use the table given here).
3. Consider the number formed by the leftover The number formed by the leftover digits is 551.
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digits in Step 2. Find the largest single digit Here we have, 8 (= 512) < 551 < 9 (= 729)
number whose cube root is less than or equal So, we take tens digit of the cube root of the given
to this leftover number. This digit is the tens number = 8
digit of the cube root. ∴ 551368 = 82
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Example 11: Find the cube root of 110592 through estimation.

Solution. Given number = 110592
= 110 592
Second group First group
The units digit of the given number of first group is 2.
So, the units digit of the cube root of the given number is 8.

The number formed by the left over digits is 110.
Since, 4 < 110 < 5 3
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So, the tens digit of the cube root of the original number = 4
Hence, the cube root of 110592, i.e., 110592 = 48
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Successive Subtraction Method

We can also find the cube root of a perfect cube number by subtracting 1, 7, 19, 37, ..., i.e., 1 + n × (n – 1) × 3,
where n is the number of terms, from the given number till we get a 0. The number of times the
subtraction is carried out, gives the cube root.
Example 12: Find the cube root of 512.

Solution: 512 – 1 = 511, 511 – 7 = 504, 504 – 19 = 485, 485 – 37 = 448, 448 – 61 = 387, 387 – 91 = 296,
296 – 127 = 169, 169 – 169 = 0
Since, we have subtracted eight times to get 0, so 512 = 8.
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157 Cubes and Cube Roots