Find the value of. `root3(392) xx root3(448)`.

Answer

To find the value of \( \sqrt[3]{392} \times \sqrt[3]{448} \), we will follow these steps:

### Step 1: Factor the numbers 392 and 448
First, we need to find the prime factorization of both numbers.

**For 392:**
1. Divide by 2: \( 392 \div 2 = 196 \)
2. Divide by 2 again: \( 196 \div 2 = 98 \)
3. Divide by 2 again: \( 98 \div 2 = 49 \)
4. Now, 49 can be factored as \( 7 \times 7 \).

So, the prime factorization of 392 is:
\[ 392 = 2^3 \times 7^2 \]

**For 448:**
1. Divide by 2: \( 448 \div 2 = 224 \)
2. Divide by 2 again: \( 224 \div 2 = 112 \)
3. Divide by 2 again: \( 112 \div 2 = 56 \)
4. Divide by 2 again: \( 56 \div 2 = 28 \)
5. Divide by 2 again: \( 28 \div 2 = 14 \)
6. Divide by 2 again: \( 14 \div 2 = 7 \)

So, the prime factorization of 448 is:
\[ 448 = 2^6 \times 7^1 \]

### Step 2: Combine the cube roots
Now we can express the original expression using the factorizations we found:
\[ \sqrt[3]{392} \times \sqrt[3]{448} = \sqrt[3]{(2^3 \times 7^2)} \times \sqrt[3]{(2^6 \times 7^1)} \]

### Step 3: Use the property of cube roots
Using the property of cube roots that states \( \sqrt[3]{a} \times \sqrt[3]{b} = \sqrt[3]{a \times b} \), we can combine the two cube roots:
\[ = \sqrt[3]{(2^3 \times 7^2) \times (2^6 \times 7^1)} \]

### Step 4: Simplify the expression inside the cube root
Now, we can simplify the expression inside the cube root:
\[ = \sqrt[3]{2^{3+6} \times 7^{2+1}} = \sqrt[3]{2^9 \times 7^3} \]

### Step 5: Calculate the cube root
Now we can take the cube root:
\[ = \sqrt[3]{(2^3)^3 \times (7^1)^3} = 2^3 \times 7^1 = 8 \times 7 = 56 \]

### Final Answer
Thus, the value of \( \sqrt[3]{392} \times \sqrt[3]{448} \) is:
\[ \boxed{56} \]