`[root(3)(2)xxsqrt(2)xxroot(3)(3)xxsqrt(3)]` is equal to

Options

A. `6^(5)`
B. `6^(5//6)`
C. `6`
D. None of these

Answer

To solve the expression \([ \sqrt[3]{2} \times \sqrt{2} \times \sqrt[3]{3} \times \sqrt{3} ]\), we will break it down step by step.

### Step 1: Rewrite the roots in terms of exponents
We can express the roots using fractional exponents:
- \(\sqrt[3]{2} = 2^{1/3}\)
- \(\sqrt{2} = 2^{1/2}\)
- \(\sqrt[3]{3} = 3^{1/3}\)
- \(\sqrt{3} = 3^{1/2}\)

So, we rewrite the expression as:
\[ 2^{1/3} \times 2^{1/2} \times 3^{1/3} \times 3^{1/2} \]

### Step 2: Combine the exponents for the same bases
Using the property of exponents that states \(a^m \times a^n = a^{m+n}\), we can combine the terms for base 2 and base 3:
- For base 2:
\[ 2^{1/3} \times 2^{1/2} = 2^{1/3 + 1/2} \]
To add the fractions, we need a common denominator. The least common multiple of 3 and 2 is 6:
\[ 1/3 = 2/6 \quad \text{and} \quad 1/2 = 3/6 \]
Thus,
\[ 1/3 + 1/2 = 2/6 + 3/6 = 5/6 \]
So, we have:
\[ 2^{1/3} \times 2^{1/2} = 2^{5/6} \]

- For base 3:
\[ 3^{1/3} \times 3^{1/2} = 3^{1/3 + 1/2} = 3^{5/6} \]

### Step 3: Combine the results
Now we can combine the results for both bases:
\[ 2^{5/6} \times 3^{5/6} \]
Using the property \(a^m \times b^m = (a \times b)^m\), we can factor this:
\[ (2 \times 3)^{5/6} = 6^{5/6} \]

### Final Answer
Thus, the expression \([ \sqrt[3]{2} \times \sqrt{2} \times \sqrt[3]{3} \times \sqrt{3} ]\) simplifies to:
\[ 6^{5/6} \]

---