The product `root3(2).root4(2). root12(32)` equal to

Options

A. `sqrt2`
B. 2
C. `root12(2)`
D. `root12(32)`

Explanation

To solve the expression \( \sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32} \), we will use the properties of exponents and roots.

### Step 1: Rewrite the roots in exponent form
We can express the roots in terms of exponents:
- \( \sqrt[3]{2} = 2^{1/3} \)
- \( \sqrt[4]{2} = 2^{1/4} \)
- \( \sqrt[12]{32} = \sqrt[12]{2^5} = (2^5)^{1/12} = 2^{5/12} \)

### Step 2: Combine the exponents
Now we can combine these exponents:
\[ \sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32} = 2^{1/3} \cdot 2^{1/4} \cdot 2^{5/12} \]
Using the property of exponents that states \( a^m \cdot a^n = a^{m+n} \), we can add the exponents:
\[ = 2^{(1/3 + 1/4 + 5/12)} \]

### Step 3: Find a common denominator
To add the fractions \( \frac{1}{3} \), \( \frac{1}{4} \), and \( \frac{5}{12} \), we need a common denominator. The least common multiple of 3, 4, and 12 is 12.

Convert each fraction:
- \( \frac{1}{3} = \frac{4}{12} \)
- \( \frac{1}{4} = \frac{3}{12} \)
- \( \frac{5}{12} = \frac{5}{12} \)

Now we can add them:
\[ \frac{4}{12} + \frac{3}{12} + \frac{5}{12} = \frac{4 + 3 + 5}{12} = \frac{12}{12} = 1 \]

### Step 4: Substitute back into the exponent
Now we substitute back into the exponent:
\[ 2^{(1/3 + 1/4 + 5/12)} = 2^{1} \]

### Step 5: Final answer
Thus, the product \( \sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32} \) equals:
\[ \boxed{2} \]