Find the cube roots of

Options

A. `-343`
B. `-4913`
C. `10648`
D. n/a

Answer

To find the cube roots of the numbers provided, we will follow a systematic approach. Let's break down the solution step by step.

### Step 1: Find the cube root of -343
1. **Identify the number**: We need to find the cube root of -343.
2. **Understand cube roots of negative numbers**: The cube root of a negative number is also negative. For example, \((-1)^3 = -1\).
3. **Find the cube root of 343**:
- We can factor 343.
- 343 can be divided by 7:
- \(343 \div 7 = 49\)
- \(49 \div 7 = 7\)
- \(7 \div 7 = 1\)
- Thus, \(343 = 7 \times 7 \times 7 = 7^3\).
4. **Calculate the cube root**:
- The cube root of \(343\) is \(7\).
- Therefore, the cube root of \(-343\) is \(-7\).

### Step 2: Find the cube root of -4913
1. **Identify the number**: We need to find the cube root of -4913.
2. **Understand cube roots of negative numbers**: Again, the cube root of a negative number is negative.
3. **Find the cube root of 4913**:
- We can factor 4913.
- 4913 can be divided by 17:
- \(4913 \div 17 = 289\)
- \(289\) can be divided by \(17\) again:
- \(289 \div 17 = 17\)
- \(17 \div 17 = 1\)
- Thus, \(4913 = 17 \times 17 \times 17 = 17^3\).
4. **Calculate the cube root**:
- The cube root of \(4913\) is \(17\).
- Therefore, the cube root of \(-4913\) is \(-17\).

### Step 3: Find the cube root of 10648
1. **Identify the number**: We need to find the cube root of 10648.
2. **Find the factors**:
- 10648 is even, so we can divide by 2:
- \(10648 \div 2 = 5324\)
- \(5324 \div 2 = 2662\)
- \(2662 \div 2 = 1331\)
- Now, we need to factor \(1331\):
- \(1331\) can be divided by \(11\):
- \(1331 \div 11 = 121\)
- \(121 \div 11 = 11\)
- \(11 \div 11 = 1\)
- Thus, \(10648 = 2^3 \times 11^3\).
3. **Calculate the cube root**:
- The cube root of \(10648\) is \(2 \times 11 = 22\).

### Final Answers
- The cube root of \(-343\) is \(-7\).
- The cube root of \(-4913\) is \(-17\).
- The cube root of \(10648\) is \(22\).