The greatest number among `root3(2)`, `sqrt(3)`, `root3(5)` and `1.5` is

Options

A. `root3(2)`
B. `root3(5)`
C. `sqrt(3)`
D. `1.5`

Answer

To find the greatest number among \( \sqrt[3]{2} \), \( \sqrt{3} \), \( \sqrt[3]{5} \), and \( 1.5 \), we can compare these numbers by expressing them in a common form. Let's follow the steps to solve this:

### Step 1: Express each number in exponential form
1. **Cube Root of 2**:
\[ \sqrt[3]{2} = 2^{1/3} \]

2. **Square Root of 3**:
\[ \sqrt{3} = 3^{1/2} \]

3. **Cube Root of 5**:
\[ \sqrt[3]{5} = 5^{1/3} \]

4. **1.5**:
\[ 1.5 = \frac{3}{2} = 3^{1} \cdot 2^{-1} \]

### Step 2: Find a common exponent
To compare these numbers effectively, we can express them with a common denominator for the exponents. The least common multiple of the denominators (3, 2, and 1) is 6. Thus, we will express each number with an exponent of 6.

1. **Cube Root of 2**:
\[ \sqrt[3]{2} = 2^{1/3} = 2^{2/6} = \sqrt[6]{2^2} = \sqrt[6]{4} \]

2. **Square Root of 3**:
\[ \sqrt{3} = 3^{1/2} = 3^{3/6} = \sqrt[6]{3^3} = \sqrt[6]{27} \]

3. **Cube Root of 5**:
\[ \sqrt[3]{5} = 5^{1/3} = 5^{2/6} = \sqrt[6]{5^2} = \sqrt[6]{25} \]

4. **1.5**:
\[ 1.5 = \frac{3}{2} = 3^{1} \cdot 2^{-1} = 3^{6/6} \cdot 2^{-6/6} = \sqrt[6]{\frac{3^6}{2^6}} = \sqrt[6]{729/64} \]

### Step 3: Compare the values
Now we have:
- \( \sqrt[6]{4} \)
- \( \sqrt[6]{27} \)
- \( \sqrt[6]{25} \)
- \( \sqrt[6]{\frac{729}{64}} \)

Next, we can compare the bases inside the radicals:
- \( 4 = 2^2 \)
- \( 27 = 3^3 \)
- \( 25 = 5^2 \)
- \( \frac{729}{64} = \frac{3^6}{2^6} \)

### Step 4: Determine the greatest value
Among the bases:
- \( 4 \) (from \( \sqrt[6]{4} \))
- \( 27 \) (from \( \sqrt[6]{27} \))
- \( 25 \) (from \( \sqrt[6]{25} \))
- \( \frac{729}{64} \) (from \( \sqrt[6]{\frac{729}{64}} \))

Clearly, \( 27 \) is the largest base. Therefore, the greatest number among \( \sqrt[3]{2} \), \( \sqrt{3} \), \( \sqrt[3]{5} \), and \( 1.5 \) is:
\[ \sqrt{3} \]

### Summary
The greatest number among \( \sqrt[3]{2} \), \( \sqrt{3} \), \( \sqrt[3]{5} \), and \( 1.5 \) is \( \sqrt{3} \).