Find the cube-roots of :
`-2744000`

Answer

To find the cube root of \(-2744000\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number**: Let \( A = -2744000 \).

2. **Set up the cube root expression**: We need to find \( A^{1/3} \), which is the same as finding \( (-2744000)^{1/3} \).

3. **Separate the negative sign**: Since the cube of a negative number is negative, we can take the negative sign outside:
\[ A^{1/3} = - (2744000)^{1/3} \]

4. **Factor the positive part**: Now we need to express \( 2744000 \) in a form that makes it easier to find the cube root. We can factor \( 2744000 \) as follows:
\[ 2744000 = 2744 \times 1000 \]

5. **Rewrite \( 1000 \)**: Recognize that \( 1000 = 10^3 \):
\[ 2744000 = 2744 \times 10^3 \]

6. **Find the cube root of \( 1000 \)**: The cube root of \( 10^3 \) is \( 10 \):
\[ (10^3)^{1/3} = 10 \]

7. **Factor \( 2744 \)**: Now we need to factor \( 2744 \). We can find that:
\[ 2744 = 2^3 \times 7^3 \]
(This can be verified by performing prime factorization.)

8. **Combine the factors**: Now we can express \( 2744000 \) as:
\[ 2744000 = (2^3 \times 7^3) \times (10^3) \]

9. **Apply the cube root**: Now we can take the cube root of the entire expression:
\[ (2^3 \times 7^3 \times 10^3)^{1/3} = 2 \times 7 \times 10 \]

10. **Calculate the product**:
\[ 2 \times 7 = 14 \]
\[ 14 \times 10 = 140 \]

11. **Include the negative sign**: Since we factored out a negative sign earlier, we have:
\[ A^{1/3} = -140 \]

### Final Answer:
The cube root of \(-2744000\) is \(-140\).

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