Which of the following are not perfect cubes?

• A. 216
• B. 128
• C. 1000
• D. 100
• E. 46656

To determine which of the numbers listed are not perfect cubes, we first need to understand what a perfect cube is. A perfect cube is a number that can be expressed as the cube of an integer. In other words, if you can find an integer \( n \) such that \( n^3 \) equals the number in question, then that number is a perfect cube. Let's analyze each option one by one.

Evaluating Each Option

• A. 216: This number is a perfect cube because \( 6^3 = 216 \).
• B. 128: This is also a perfect cube since \( 2^3 = 8 \) and \( 8 \times 16 = 128 \), or more directly, \( 4^3 = 64 \) and \( 5^3 = 125 \), so it falls between those two. However, \( 128 \) can be expressed as \( 2^7 \), which is not a perfect cube.
• C. 1000: This is a perfect cube because \( 10^3 = 1000 \).
• D. 100: This number is not a perfect cube. The cube roots of integers around 100 are \( 4^3 = 64 \) and \( 5^3 = 125 \), and 100 does not equal \( n^3 \) for any integer \( n \).
• E. 46656: This is a perfect cube since \( 36^3 = 46656 \).

Summary of Findings

From our analysis, we can conclude that the numbers that are not perfect cubes from the list provided are:

• 128
• 100

Thus, the answer to your question is that both 128 and 100 are not perfect cubes. Understanding the properties of cubes and how to calculate them can help in identifying such numbers in the future.