which of the following are cubes of :
an even number 216,512,729,1000,3375,13824

Answer

To determine which of the given numbers are cubes of even numbers, we will follow these steps:

### Step 1: Understand the concept of cubes
A cube of a number \( n \) is calculated as \( n \times n \times n \) or \( n^3 \). For example, the cube of 2 is \( 2^3 = 2 \times 2 \times 2 = 8 \).

### Step 2: Identify the given numbers
The numbers we need to analyze are:
- 216
- 512
- 729
- 1000
- 3375
- 13824

### Step 3: Check which numbers are even
Even numbers are those that are divisible by 2. From the list, we can identify:
- 216 (even)
- 512 (even)
- 729 (odd)
- 1000 (even)
- 3375 (odd)
- 13824 (even)

The even numbers from the list are: 216, 512, 1000, and 13824.

### Step 4: Calculate the cube roots of the even numbers
Next, we will find the cube root of each even number to check if they are indeed cubes of even numbers.

1. **216**:
- Cube root: \( \sqrt[3]{216} = 6 \) (even)

2. **512**:
- Cube root: \( \sqrt[3]{512} = 8 \) (even)

3. **1000**:
- Cube root: \( \sqrt[3]{1000} = 10 \) (even)

4. **13824**:
- Cube root: \( \sqrt[3]{13824} = 24 \) (even)

### Step 5: Conclusion
The numbers that are cubes of even numbers from the given list are:
- 216
- 512
- 1000
- 13824

### Summary
Thus, the cubes of even numbers among the given options are 216, 512, 1000, and 13824.

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