Greatest among the numbers `root3(9)`, `sqrt(3)`, `root4(16)`, `root6(80)` is

Options

A. `root3(9)`
B. `sqrt(3)`
C. `root4(16)`
D. `root6(80)`

Answer

To find the greatest among the numbers \( \sqrt[3]{9} \), \( \sqrt{3} \), \( \sqrt[4]{16} \), and \( \sqrt[6]{80} \), we will convert each of these roots into powers of 3, so we can compare them easily.

### Step 1: Convert each root into exponential form

1. **Convert \( \sqrt[3]{9} \)**:
\[ \sqrt[3]{9} = 9^{1/3} = (3^2)^{1/3} = 3^{2/3} \]

2. **Convert \( \sqrt{3} \)**:
\[ \sqrt{3} = 3^{1/2} \]

3. **Convert \( \sqrt[4]{16} \)**:
\[ \sqrt[4]{16} = 16^{1/4} = (4^2)^{1/4} = 4^{1/2} = 2^{1/2} \cdot 2^{1/2} = 2^{1} = 2 \]
Since \( 2 = 3^{\log_3{2}} \), we can keep it as \( 2^{1} \) for now.

4. **Convert \( \sqrt[6]{80} \)**:
\[ \sqrt[6]{80} = 80^{1/6} = (16 \cdot 5)^{1/6} = (4^2 \cdot 5)^{1/6} = (2^4 \cdot 5)^{1/6} = 2^{4/6} \cdot 5^{1/6} = 2^{2/3} \cdot 5^{1/6} \]

### Step 2: Find a common exponent to compare

To compare these numbers, we can find a common base. The least common multiple of the denominators of the exponents (3, 2, 1, 6) is 6. We will convert all the expressions to have a denominator of 6.

1. **For \( 3^{2/3} \)**:
\[ 3^{2/3} = 3^{4/6} \]

2. **For \( 3^{1/2} \)**:
\[ 3^{1/2} = 3^{3/6} \]

3. **For \( 2 \)**:
\[ 2 = 2^{1} = 2^{6/6} = 3^{\log_3{2}} \text{ (but we will keep it as is for comparison)} \]

4. **For \( 2^{2/3} \cdot 5^{1/6} \)**:
\[ 2^{2/3} = 2^{4/6} \quad \text{and} \quad 5^{1/6} = 5^{1/6} \]

### Step 3: Calculate the values

Now we can calculate the approximate values of these expressions:

1. **Calculate \( 3^{4/6} \)**:
\[ 3^{4/6} \approx 3^{0.67} \approx 2.08 \]

2. **Calculate \( 3^{3/6} \)**:
\[ 3^{3/6} \approx 3^{0.5} \approx 1.73 \]

3. **Calculate \( 2^{6/6} \)**:
\[ 2^{1} = 2 \]

4. **Calculate \( 2^{4/6} \cdot 5^{1/6} \)**:
\[ 2^{4/6} \approx 1.587 \quad \text{and} \quad 5^{1/6} \approx 1.348 \]
\[ 2^{4/6} \cdot 5^{1/6} \approx 1.587 \cdot 1.348 \approx 2.14 \]

### Step 4: Compare the values

Now we compare the approximate values:
- \( \sqrt[3]{9} \approx 2.08 \)
- \( \sqrt{3} \approx 1.73 \)
- \( \sqrt[4]{16} = 2 \)
- \( \sqrt[6]{80} \approx 2.14 \)

### Conclusion

The greatest among the numbers \( \sqrt[3]{9} \), \( \sqrt{3} \), \( \sqrt[4]{16} \), and \( \sqrt[6]{80} \) is \( \sqrt[6]{80} \).