'""^(3)sqrt(2) xx ""^(4)sqrt(2) xx ""^(12)sqrt(32)= ?`

Options

A. 2
B. `sqrt(2)`
C. `2sqrt(2)`
D. `4sqrt(2)`

Answer

To solve the expression \( \sqrt[3]{2} \times \sqrt[4]{2} \times \sqrt[12]{32} \), we can follow these steps:

### Step 1: Rewrite the roots in exponential form
We can express the roots in terms of exponents:
\[ \sqrt[3]{2} = 2^{1/3}, \quad \sqrt[4]{2} = 2^{1/4}, \quad \text{and} \quad \sqrt[12]{32} = 32^{1/12} \]

### Step 2: Simplify \( \sqrt[12]{32} \)
Next, we need to express 32 as a power of 2:
\[ 32 = 2^5 \]
Thus,
\[ \sqrt[12]{32} = (2^5)^{1/12} = 2^{5/12} \]

### Step 3: Combine all the terms
Now we can rewrite the entire expression:
\[ 2^{1/3} \times 2^{1/4} \times 2^{5/12} \]

### Step 4: Add the exponents
Since the bases are the same, we can add the exponents:
\[ 2^{1/3 + 1/4 + 5/12} \]

### Step 5: Find a common denominator
The common denominator for \(3\), \(4\), and \(12\) is \(12\). We convert each fraction:
\[ 1/3 = 4/12, \quad 1/4 = 3/12, \quad 5/12 = 5/12 \]
Now we can add them:
\[ 4/12 + 3/12 + 5/12 = \frac{4 + 3 + 5}{12} = \frac{12}{12} = 1 \]

### Step 6: Write the final result
Now we can write the final expression:
\[ 2^{1} = 2 \]

Thus, the final answer is:
\[ \boxed{2} \]

---