Question

Which of the following are not perfect cubes? Give the reasons in support of your answer.
A. 648
B. 729
C. 8640
D. 8000

Answer 1

Hint: This problem involves the concept of cubes. A perfect cube is a number whose cube root is an integer. In other words, it is a number which can be expressed as the cube of an integer. We will use hit and trial method to check if the numbers are perfect cubes or not. We will find the numbers which are close to the cube roots of the given numbers. The cube of a number A can be written as-
${A^3} = A \times A \times A$

Complete step-by-step answer:
We will find the numbers which have a close to each of the four numbers above. Then we will successively increase the number to see if they are a perfect cube or not.
In option A, the number given is 648. First we will find a perfect cube close to this. When we use the number 8, the cube is 512. That is-
${8^3} = 8 \times 8 \times 8 = 512$
This is less than the number 648, so we will increase the number by 1, that is 9. When we find the cube of 9, it is equal to 729. That is-
${9^3} = 9 \times 9 \times 9 = 729$
We can see that the number 648 lies between these two successive perfect cubes. So it is not a perfect cube.
In option B, the number given is 729. We know that this number is a cube of 9. Therefore 729 is a perfect cube.
In option C, the given number is 8640. First we will find a perfect cube close to this number. It is a large number, so we will use the number 20. The cube of the number 20 is 8000. That is-
${20^3} = 20 \times 20 \times 20 = 8000$
This perfect cube is smaller than 8640. So, we will consider the next perfect cube which is-
${21^3} = 21 \times 21 \times 21 = 9261$
We can see that the number 8640 lies between these two successive perfect cubes. So it is not a perfect cube.
In option D, the number is 8000. We can clearly get a hint that it is a cube of a number who has 0 as its last digit. So, we will consider the number 20 which is-
${20^3} = 20 \times 20 \times 20 = 8000$
Hence, it is a perfect cube.

From the above options, we can see that 648 and 8640 are not perfect cubes. Hence, the correct options are A and C.

Note:In such types of questions, hit and trial is an appropriate method. This is because we do not have a direct method to solve a cube root of a given number. A common mistake is that students may mark the options which are perfect cubes, instead of the other way round. Also, students in a hurry may check the number for perfect squares instead of perfect cubes. These kinds of mistakes should be avoided.