Find the cube-roots of :
343

Answer

To find the cube root of 343, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the number**: We need to find the cube root of 343. Let's denote this number as \( A \).
\[ A = 343 \]

2. **Understand the cube root**: The cube root of a number \( A \) can be expressed mathematically as:
\[ \sqrt[3]{A} = A^{\frac{1}{3}} \]
Therefore, we need to calculate:
\[ \sqrt[3]{343} = 343^{\frac{1}{3}} \]

3. **Factor the number**: Next, we can express 343 as a power of a prime number. We can check if 343 is a perfect cube. We can do this by factorization:
\[ 343 = 7 \times 7 \times 7 = 7^3 \]

4. **Apply the cube root formula**: Now that we have expressed 343 as \( 7^3 \), we can substitute this back into our cube root expression:
\[ \sqrt[3]{343} = \sqrt[3]{7^3} \]

5. **Simplify using exponent rules**: We can use the property of exponents that states \( (a^m)^n = a^{m \cdot n} \):
\[ \sqrt[3]{7^3} = 7^{3 \cdot \frac{1}{3}} = 7^1 = 7 \]

6. **Conclusion**: Therefore, the cube root of 343 is:
\[ \sqrt[3]{343} = 7 \]

### Final Answer:
The cube root of 343 is \( 7 \).

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