Divide `12xx4^(1//3)" by "3sqrt2`.

Answer

To solve the problem of dividing \( 12 \times 4^{\frac{1}{3}} \) by \( 3\sqrt{2} \), we will follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the division as follows:

\[ \frac{12 \times 4^{\frac{1}{3}}}{3\sqrt{2}} \]

### Step 2: Simplify the denominator
Next, we rewrite \( \sqrt{2} \) in terms of exponents:

\[ \sqrt{2} = 2^{\frac{1}{2}} \]

Thus, the denominator becomes:

\[ 3\sqrt{2} = 3 \times 2^{\frac{1}{2}} \]

### Step 3: Substitute the denominator back into the expression
Now we can rewrite the entire expression:

\[ \frac{12 \times 4^{\frac{1}{3}}}{3 \times 2^{\frac{1}{2}}} \]

### Step 4: Factor out the constants
We can factor \( 12 \) as \( 3 \times 4 \):

\[ \frac{3 \times 4 \times 4^{\frac{1}{3}}}{3 \times 2^{\frac{1}{2}}} \]

### Step 5: Cancel out the common terms
Now we can cancel the \( 3 \) in the numerator and denominator:

\[ \frac{4 \times 4^{\frac{1}{3}}}{2^{\frac{1}{2}}} \]

### Step 6: Rewrite \( 4 \) in terms of base \( 2 \)
We know that \( 4 = 2^2 \), so we can rewrite the numerator:

\[ \frac{2^2 \times 4^{\frac{1}{3}}}{2^{\frac{1}{2}}} \]

### Step 7: Rewrite \( 4^{\frac{1}{3}} \)
Now, we can express \( 4^{\frac{1}{3}} \) as \( (2^2)^{\frac{1}{3}} = 2^{\frac{2}{3}} \):

\[ \frac{2^2 \times 2^{\frac{2}{3}}}{2^{\frac{1}{2}}} \]

### Step 8: Combine the powers of \( 2 \)
Using the property of exponents \( a^m \times a^n = a^{m+n} \):

\[ \frac{2^{2 + \frac{2}{3}}}{2^{\frac{1}{2}}} \]

### Step 9: Simplify the exponent in the numerator
Now we need to add the exponents in the numerator:

\[ 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3} \]

So we have:

\[ \frac{2^{\frac{8}{3}}}{2^{\frac{1}{2}}} \]

### Step 10: Subtract the exponents
Now we can subtract the exponents since the bases are the same:

\[ 2^{\frac{8}{3} - \frac{1}{2}} \]

### Step 11: Find a common denominator
To perform the subtraction, we need a common denominator. The least common multiple of 3 and 2 is 6:

\[ \frac{8}{3} = \frac{16}{6}, \quad \frac{1}{2} = \frac{3}{6} \]

Now we can subtract:

\[ \frac{16}{6} - \frac{3}{6} = \frac{13}{6} \]

### Step 12: Final result
Thus, we have:

\[ 2^{\frac{13}{6}} \]

This is the final answer.

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