Which of the given numbers is greatest ?
`6root3(5), root3(2),2root3(130),root3(900)`.

Answer

To find which of the given numbers is the greatest among \(6\sqrt[3]{5}\), \(\sqrt[3]{2}\), \(2\sqrt[3]{130}\), and \(\sqrt[3]{900}\), we can compare them step by step.

### Step 1: Rewrite the expressions
We can express each number in terms of a common base for easier comparison.

1. \( A = 6\sqrt[3]{5} = 6 \cdot 5^{1/3} \)
2. \( B = \sqrt[3]{2} = 2^{1/3} \)
3. \( C = 2\sqrt[3]{130} = 2 \cdot 130^{1/3} \)
4. \( D = \sqrt[3]{900} = 900^{1/3} \)

### Step 2: Simplify the expressions
Next, we can simplify \(C\) and \(D\):

- For \(C\):
\[ C = 2 \cdot (2 \cdot 65)^{1/3} = 2^{1 + 1/3} \cdot 65^{1/3} = 2^{4/3} \cdot 65^{1/3} \]

- For \(D\):
\[ D = (900)^{1/3} = (9 \cdot 100)^{1/3} = (3^2 \cdot 10^2)^{1/3} = 3^{2/3} \cdot 10^{2/3} \]

### Step 3: Compare the values
Now we can compare the values of \(A\), \(B\), \(C\), and \(D\):

1. **Comparing \(A\) and \(B\)**:
\[ A = 6 \cdot 5^{1/3} \quad \text{and} \quad B = 2^{1/3} \]
Since \(6 > 2\) and \(5^{1/3} > 1\), we can conclude that \(A > B\).

2. **Comparing \(A\) and \(C\)**:
\[ A = 6 \cdot 5^{1/3} \quad \text{and} \quad C = 2^{4/3} \cdot 65^{1/3} \]
To compare, we can evaluate both expressions numerically or simplify further.

3. **Comparing \(C\) and \(D\)**:
\[ C = 2^{4/3} \cdot 65^{1/3} \quad \text{and} \quad D = 3^{2/3} \cdot 10^{2/3} \]
Again, we can evaluate both expressions numerically or simplify further.

### Step 4: Numerical approximation
To find the greatest number, we can calculate the numerical values of each expression:

1. \(A = 6 \cdot 5^{1/3} \approx 6 \cdot 1.71 \approx 10.26\)
2. \(B = 2^{1/3} \approx 1.26\)
3. \(C = 2^{4/3} \cdot 65^{1/3} \approx 2.52 \cdot 4.02 \approx 10.14\)
4. \(D = 3^{2/3} \cdot 10^{2/3} \approx 2.08 \cdot 4.64 \approx 9.66\)

### Conclusion
From the calculations, we find:
- \(A \approx 10.26\)
- \(B \approx 1.26\)
- \(C \approx 10.14\)
- \(D \approx 9.66\)

Thus, the greatest number is \(A = 6\sqrt[3]{5}\).