Find the smallest number which must be subtracted from the following to make them perfect cubes .What are the corresponding cube roots?

Options

A. 350
B. 833
C. 1400
D. 1730

Answer

To solve the problem of finding the smallest number that must be subtracted from given numbers to make them perfect cubes, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the given numbers**: The numbers we need to work with are 350, 833, 1400, and 1730.

2. **Find the cube roots of nearby perfect cubes**: We will list the perfect cubes around the given numbers:
- The cubes of integers from 1 to 12 are:
- \(1^3 = 1\)
- \(2^3 = 8\)
- \(3^3 = 27\)
- \(4^3 = 64\)
- \(5^3 = 125\)
- \(6^3 = 216\)
- \(7^3 = 343\)
- \(8^3 = 512\)
- \(9^3 = 729\)
- \(10^3 = 1000\)
- \(11^3 = 1331\)
- \(12^3 = 1728\)

3. **Determine the nearest perfect cube for each number**:
- For **350**: The nearest perfect cube is \(343\) (which is \(7^3\)).
- For **833**: The nearest perfect cube is \(729\) (which is \(9^3\)).
- For **1400**: The nearest perfect cube is \(1331\) (which is \(11^3\)).
- For **1730**: The nearest perfect cube is \(1728\) (which is \(12^3\)).

4. **Calculate the difference for each number**:
- For **350**:
\[ 350 - 343 = 7 \]
- For **833**:
\[ 833 - 729 = 104 \]
- For **1400**:
\[ 1400 - 1331 = 69 \]
- For **1730**:
\[ 1730 - 1728 = 2 \]

5. **Summarize the results**:
- From **350**, subtract **7** to get **343** (cube root is **7**).
- From **833**, subtract **104** to get **729** (cube root is **9**).
- From **1400**, subtract **69** to get **1331** (cube root is **11**).
- From **1730**, subtract **2** to get **1728** (cube root is **12**).

### Final Results:
- For **350**: Subtract **7**, cube root is **7**.
- For **833**: Subtract **104**, cube root is **9**.
- For **1400**: Subtract **69**, cube root is **11**.
- For **1730**: Subtract **2**, cube root is **12**.