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# Estimate the cube root of 110592.A. 48B. 52C. 46D. 54

Last updated date: 13th Jun 2024
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Hint: Numbers that are obtained when a number is multiplied by itself three times are known as cube numbers.
If $m={{n}^{3}}$ , then $m$ is the cube of $n$, and $n$ is the cube root of $m$ .
A number whose units digit is 8 has a cube whose units digit is 2.
If a number has $n$ number of zeros at its end then it's cube will have $3n$ number of zeros at its end.
Observe that ${{40}^{3}}=64000$ and ${{50}^{3}}=125000$ .

Since the number 110592 lies between ${{40}^{3}}=64000$ and ${{50}^{3}}=125000$ , its cube root must be between 40 and 50.
Also, the unit digit of 110592 is 2, therefore, the unit digit of its cube root must be 8.
So, the cube root of 110592 will be 48, provided 110592 is a perfect cube.
We check that $48\times 48=2304$ and $2304\times 48=110592$ .
Therefore, the correct answer is A. 48.

Note: Cube numbers are also known as perfect cubes.
Cubes of the numbers with units digit 1, 4, 5, 6 and 9 are the numbers ending in the same units digit.
A number whose units digit is 2 has a cube whose units digit is 8 and vice versa.
A number whose units digit is 3 has a cube whose units digit is 7 and vice versa.
Some other Properties of Cube Numbers:
If a number has n number of zeros at its end then it's cube will have 3n number of zeros at its end.
The cube of an even number is always even and the cube of an odd number is always odd.
When a perfect cube or cube number is prime factorized, its factors can be grouped into triplets; groups of 3 identical primes.