Find the cube-roots of :
`343/512`

Answer

To find the cube root of \( \frac{343}{512} \), we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 343
- We start by finding the prime factors of 343.
- The smallest prime number that divides 343 is 7.
- Dividing 343 by 7:
\[ 343 \div 7 = 49 \]
- Now, we factor 49:
\[ 49 \div 7 = 7 \]
- Finally, we have:
\[ 7 \div 7 = 1 \]
- So, the prime factorization of 343 is:
\[ 343 = 7 \times 7 \times 7 = 7^3 \]

### Step 2: Prime Factorization of 512
- Next, we find the prime factors of 512.
- The smallest prime number that divides 512 is 2.
- Dividing 512 by 2 repeatedly:
\[ 512 \div 2 = 256 \]
\[ 256 \div 2 = 128 \]
\[ 128 \div 2 = 64 \]
\[ 64 \div 2 = 32 \]
\[ 32 \div 2 = 16 \]
\[ 16 \div 2 = 8 \]
\[ 8 \div 2 = 4 \]
\[ 4 \div 2 = 2 \]
\[ 2 \div 2 = 1 \]
- Thus, the prime factorization of 512 is:
\[ 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 \]

### Step 3: Write the Fraction with Prime Factors
- Now we can express \( \frac{343}{512} \) using the prime factorizations:
\[ \frac{343}{512} = \frac{7^3}{2^9} \]

### Step 4: Find the Cube Root
- To find the cube root of \( \frac{7^3}{2^9} \), we apply the cube root to both the numerator and the denominator:
\[ \sqrt[3]{\frac{7^3}{2^9}} = \frac{\sqrt[3]{7^3}}{\sqrt[3]{2^9}} = \frac{7}{2^3} \]
- Since \( 2^3 = 8 \), we have:
\[ \frac{7}{8} \]

### Final Answer
Thus, the cube root of \( \frac{343}{512} \) is:
\[ \frac{7}{8} \]

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