Find the cube root of `3375/2744`

Answer

To find the cube root of \( \frac{3375}{2744} \), we will follow these steps:

### Step 1: Prime Factorization of 3375
We start by finding the prime factorization of 3375.

1. Divide by 3:
- \( 3375 \div 3 = 1125 \)
2. Divide by 3 again:
- \( 1125 \div 3 = 375 \)
3. Divide by 3 again:
- \( 375 \div 3 = 125 \)
4. Divide by 5:
- \( 125 \div 5 = 25 \)
5. Divide by 5 again:
- \( 25 \div 5 = 5 \)
6. Divide by 5 again:
- \( 5 \div 5 = 1 \)

Thus, the prime factorization of 3375 is:
\[ 3375 = 3^3 \times 5^3 \]

### Step 2: Prime Factorization of 2744
Next, we find the prime factorization of 2744.

1. Divide by 2:
- \( 2744 \div 2 = 1372 \)
2. Divide by 2 again:
- \( 1372 \div 2 = 686 \)
3. Divide by 2 again:
- \( 686 \div 2 = 343 \)
4. Divide by 7:
- \( 343 \div 7 = 49 \)
5. Divide by 7 again:
- \( 49 \div 7 = 7 \)
6. Divide by 7 again:
- \( 7 \div 7 = 1 \)

Thus, the prime factorization of 2744 is:
\[ 2744 = 2^3 \times 7^3 \]

### Step 3: Formulate the Expression
Now we can express \( \frac{3375}{2744} \) using the prime factorizations we found:
\[ \frac{3375}{2744} = \frac{3^3 \times 5^3}{2^3 \times 7^3} \]

### Step 4: Simplify the Expression
This can be simplified as:
\[ \frac{3375}{2744} = \frac{3^3 \times 5^3}{2^3 \times 7^3} = \frac{3^3}{2^3} \times \frac{5^3}{7^3} \]

### Step 5: Find the Cube Root
Now we can find the cube root of the entire expression:
\[ \sqrt[3]{\frac{3375}{2744}} = \sqrt[3]{\frac{3^3 \times 5^3}{2^3 \times 7^3}} = \frac{\sqrt[3]{3^3} \times \sqrt[3]{5^3}}{\sqrt[3]{2^3} \times \sqrt[3]{7^3}} \]

This simplifies to:
\[ \frac{3 \times 5}{2 \times 7} = \frac{15}{14} \]

### Final Answer
Thus, the cube root of \( \frac{3375}{2744} \) is:
\[ \frac{15}{14} \]

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