To determine if a number is a perfect cube, we need to check if there exists an integer [tex]\( n \)[/tex] such that [tex]\( n^3 \)[/tex] equals the given number. Below is the step-by-step process for each number provided in the question.
1. 216
We check if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^3 = 216 \)[/tex].
- [tex]\( 6^3 = 216 \)[/tex].
Hence, 216 is a perfect cube.
2. 46656
We check if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^3 = 46656 \)[/tex].
- [tex]\( 36^3 = 46656 \)[/tex].
Hence, 46656 is a perfect cube.
3. 128
We check if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^3 = 128 \)[/tex].
- [tex]\( 5^3 = 125 \)[/tex] and [tex]\( 6^3 = 216 \)[/tex]; 128 does not fall between these cubes perfectly.
Hence, 128 is not a perfect cube.
4. 1000
We check if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^3 = 1000 \)[/tex].
- [tex]\( 10^3 = 1000 \)[/tex].
Hence, 1000 is a perfect cube.
5. 100
We check if there is an integer [tex]\( n \)[/tex] such that [tex]\( n^3 = 100 \)[/tex].
- [tex]\( 4^3 = 64 \)[/tex] and [tex]\( 5^3 = 125 \)[/tex]; 100 does not fall between these cubes perfectly.
Hence, 100 is not a perfect cube.
Therefore, the numbers that are not perfect cubes are:
(ii) 128 and (iv) 100.
