The value of `3 root(3) (2) xx 7 root(3) (6) xx 5 root(3)(18)` is

Options

A. 545
B. 500
C. 630
D. 400

Answer

To find the value of the expression \(3 \sqrt[3]{2} \times 7 \sqrt[3]{6} \times 5 \sqrt[3]{18}\), we will break it down step by step.

### Step 1: Rewrite the roots in terms of exponents
We can express the cube roots using exponents:
- \(\sqrt[3]{2} = 2^{1/3}\)
- \(\sqrt[3]{6} = 6^{1/3} = (2 \times 3)^{1/3} = 2^{1/3} \times 3^{1/3}\)
- \(\sqrt[3]{18} = 18^{1/3} = (2 \times 3^2)^{1/3} = 2^{1/3} \times 3^{2/3}\)

So, we can rewrite the expression:
\[ 3 \cdot 2^{1/3} \cdot 7 \cdot (2^{1/3} \cdot 3^{1/3}) \cdot 5 \cdot (2^{1/3} \cdot 3^{2/3}) \]

### Step 2: Combine the terms
Now, we can combine the terms:
\[ = 3 \cdot 7 \cdot 5 \cdot 2^{1/3} \cdot 2^{1/3} \cdot 2^{1/3} \cdot 3^{1/3} \cdot 3^{2/3} \]

### Step 3: Simplify the powers
Combine the powers of 2 and 3:
- For \(2\): \(2^{1/3} \cdot 2^{1/3} \cdot 2^{1/3} = 2^{1/3 + 1/3 + 1/3} = 2^{3/3} = 2^1 = 2\)
- For \(3\): \(3^{1/3} \cdot 3^{2/3} = 3^{1/3 + 2/3} = 3^{3/3} = 3^1 = 3\)

### Step 4: Calculate the coefficients
Now, calculate the coefficients:
\[ 3 \cdot 7 \cdot 5 = 105 \]

### Step 5: Combine everything
Putting it all together:
\[ = 105 \cdot 2 \cdot 3 = 105 \cdot 6 = 630 \]

Thus, the value of the expression \(3 \sqrt[3]{2} \times 7 \sqrt[3]{6} \times 5 \sqrt[3]{18}\) is \(630\).