Join the IDNLearn.com community and start exploring a world of knowledge today. Discover in-depth and trustworthy answers to all your questions from our experienced community members.
Sagot :
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.
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.
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.