Find the cube-roots of :
`3375xx512`

Answer

To find the cube roots of the number \( 3375 \times 512 \), we can follow these steps:

### Step 1: Factor the numbers
First, we need to factor both \( 3375 \) and \( 512 \).

**For \( 3375 \):**
- \( 3375 \) is divisible by \( 3 \):
- \( 3375 \div 3 = 1125 \)
- \( 1125 \) is also divisible by \( 3 \):
- \( 1125 \div 3 = 375 \)
- \( 375 \) is again divisible by \( 3 \):
- \( 375 \div 3 = 125 \)
- \( 125 \) is divisible by \( 5 \):
- \( 125 \div 5 = 25 \)
- \( 25 \) is also divisible by \( 5 \):
- \( 25 \div 5 = 5 \)
- Finally, \( 5 \div 5 = 1 \)

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

**For \( 512 \):**
- \( 512 \) is divisible by \( 2 \):
- \( 512 \div 2 = 256 \)
- \( 256 \) is also divisible by \( 2 \):
- \( 256 \div 2 = 128 \)
- \( 128 \) is again divisible by \( 2 \):
- \( 128 \div 2 = 64 \)
- \( 64 \) is divisible by \( 2 \):
- \( 64 \div 2 = 32 \)
- \( 32 \) is divisible by \( 2 \):
- \( 32 \div 2 = 16 \)
- \( 16 \) is divisible by \( 2 \):
- \( 16 \div 2 = 8 \)
- \( 8 \) is divisible by \( 2 \):
- \( 8 \div 2 = 4 \)
- \( 4 \) is divisible by \( 2 \):
- \( 4 \div 2 = 2 \)
- Finally, \( 2 \div 2 = 1 \)

So, the prime factorization of \( 512 \) is:
\[ 512 = 2^9 \]

### Step 2: Combine the factors
Now, we can combine the factors from both numbers:
\[ 3375 \times 512 = (3^3 \times 5^3) \times (2^9) = 3^3 \times 5^3 \times 2^9 \]

### Step 3: Find the cube root
To find the cube root of the combined product, we apply the cube root to each factor:
\[ \sqrt[3]{3375 \times 512} = \sqrt[3]{3^3 \times 5^3 \times 2^9} = 3^{3/3} \times 5^{3/3} \times 2^{9/3} \]
This simplifies to:
\[ 3^1 \times 5^1 \times 2^3 = 3 \times 5 \times 8 \]

### Step 4: Calculate the final result
Now we can calculate:
\[ 3 \times 5 = 15 \]
\[ 15 \times 8 = 120 \]

Thus, the cube root of \( 3375 \times 512 \) is:
\[ \boxed{120} \]