Evaluate : `root3(1372) xx root3(1458)`.

Answer

To evaluate \( \sqrt[3]{1372} \times \sqrt[3]{1458} \), we will follow these steps:

### Step 1: Prime Factorization of 1372
We start by finding the prime factorization of 1372.

1. Divide 1372 by 2:
\[ 1372 \div 2 = 686 \]
2. Divide 686 by 2:
\[ 686 \div 2 = 343 \]
3. Now, divide 343 by 7:
\[ 343 \div 7 = 49 \]
4. Divide 49 by 7:
\[ 49 \div 7 = 7 \]
5. Finally, divide 7 by 7:
\[ 7 \div 7 = 1 \]

Thus, the prime factorization of 1372 is:
\[ 1372 = 2^2 \times 7^3 \]

### Step 2: Prime Factorization of 1458
Next, we find the prime factorization of 1458.

1. Divide 1458 by 2:
\[ 1458 \div 2 = 729 \]
2. Now, divide 729 by 3:
\[ 729 \div 3 = 243 \]
3. Divide 243 by 3:
\[ 243 \div 3 = 81 \]
4. Divide 81 by 3:
\[ 81 \div 3 = 27 \]
5. Divide 27 by 3:
\[ 27 \div 3 = 9 \]
6. Divide 9 by 3:
\[ 9 \div 3 = 3 \]
7. Finally, divide 3 by 3:
\[ 3 \div 3 = 1 \]

Thus, the prime factorization of 1458 is:
\[ 1458 = 2^1 \times 3^6 \]

### Step 3: Combine the Prime Factorizations
Now we can combine the prime factorizations of both numbers:
\[ \sqrt[3]{1372} \times \sqrt[3]{1458} = \sqrt[3]{(2^2 \times 7^3) \times (2^1 \times 3^6)} \]

Combine the factors:
\[ = \sqrt[3]{2^{2+1} \times 7^3 \times 3^6} = \sqrt[3]{2^3 \times 7^3 \times 3^6} \]

### Step 4: Simplify the Expression
Now we can simplify the expression:
\[ = \sqrt[3]{(2 \times 7 \times 3^2)^3} \]
Since the cube root and the cube cancel each other out:
\[ = 2 \times 7 \times 3^2 \]

### Step 5: Calculate the Final Result
Now, calculate the final result:
\[ = 2 \times 7 \times 9 = 14 \times 9 = 126 \]

Thus, the final answer is:
\[ \sqrt[3]{1372} \times \sqrt[3]{1458} = 126 \]