Question

The largest four-digit number, which is a perfect cube is:
A. 8000
B. 9261
C. 9999
D. 8889

Answer 1

Hint: We can check which of the options is a perfect cube. As all the options are four-digit numbers, the highest number which is a perfect cube will be the correct answer. We can also check the cube of random consecutive numbers whose cubes come in the given range.

Complete step by step Answer:

Cube of a number is the number in its third power. Cube of a number n is defined as,
${n^3} = {n^2} \times n = n \times n \times n$.
A number is said to be a perfect cube; it can be expressed as the cube of an integer.
We can take the smallest number in the options and find its cube root. The smallest number in the
The option is 8000.
8000 can be factorized as follows,
$8000 = 8 \times 1000$
We know that, 8 and 1000 are cubes of 2 and 10 respectively,
$ \Rightarrow 8000 = {2^3} \times {10^3}$
By laws of exponents, we can write,
\[ \Rightarrow 8000 = {\left( {2 \times 10} \right)^3}\]
\[ \Rightarrow 8000 = {20^3}\]
Now, we can take the cube of the next number after 20 which is 21.
${\left( {21} \right)^3} = 21 \times 21 \times 21$
$ = 441 \times 21$
$ = 9261$
We can also check the cube root of the next number, which is 22
${\left( {22} \right)^3} = 22 \times 22 \times 22$
$ = 484 \times 22$
$ = 10648$
As the cube of 22 is a five-digit number, the largest four-digit number which is a perfect cube is the cube root of 21.
So, the required number is 9261.
Therefore, the correct answer is option B.

Note: Cube of a number is the number in its third power. Cube of a number n is defined as,
${n^3} = {n^2} \times n = n \times n \times n$. It can also be defined as the volume of the cube with the length of the side n. The same method we used here can be used for finding the perfect square and also a different number of digits. Taking the cube of a number will retain the sign of the number. In other words, the cube of a number can be positive or negative. For squares, the square of a number will always be positive. Numbers with negative squares will be complex numbers. We must be careful while finding the cubes as it involves multiplication of three-digit numbers.