Evaluate `root3(216 xx 343)`

Options

A. `36`
B. `45`
C. `42`
D. `49`

Answer

To evaluate \( \sqrt[3]{216 \times 343} \), we can follow these steps:

### Step 1: Factor the numbers
First, we need to factor both numbers, 216 and 343, into their prime factors.

- **For 216**:
- \( 216 = 6 \times 36 \)
- \( 36 = 6 \times 6 \)
- Therefore, \( 216 = 6 \times 6 \times 6 = 6^3 \).

- **For 343**:
- \( 343 = 7 \times 49 \)
- \( 49 = 7 \times 7 \)
- Therefore, \( 343 = 7 \times 7 \times 7 = 7^3 \).

### Step 2: Rewrite the expression
Now we can rewrite the expression \( \sqrt[3]{216 \times 343} \) using the prime factorization:
\[ \sqrt[3]{216 \times 343} = \sqrt[3]{6^3 \times 7^3} \]

### Step 3: Apply the property of cube roots
Using the property of cube roots, we can separate the cube roots:
\[ \sqrt[3]{6^3 \times 7^3} = \sqrt[3]{6^3} \times \sqrt[3]{7^3} \]

### Step 4: Simplify the cube roots
Now we can simplify each cube root:
\[ \sqrt[3]{6^3} = 6 \quad \text{and} \quad \sqrt[3]{7^3} = 7 \]

### Step 5: Multiply the results
Now we multiply the results of the cube roots:
\[ 6 \times 7 = 42 \]

### Final Answer
Thus, the value of \( \sqrt[3]{216 \times 343} \) is \( 42 \).

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