The largest number among `sqrt(2)`, `root3(3)`, `root4(4)` is

Options

A. `sqrt(2)`
B. `root3(3)`
C. root4(4)`
D. All are equal

Answer

To find the largest number among \( \sqrt{2} \), \( \sqrt[3]{3} \), and \( \sqrt[4]{4} \), we can express each of these numbers in terms of powers of 2, 3, and 4.

### Step 1: Rewrite the roots in exponential form
1. \( \sqrt{2} = 2^{1/2} \)
2. \( \sqrt[3]{3} = 3^{1/3} \)
3. \( \sqrt[4]{4} = 4^{1/4} = (2^2)^{1/4} = 2^{1/2} \)

### Step 2: Find a common base for comparison
To compare these values, we can convert them into a common base. Since \( 2 \) is a common base for \( \sqrt{2} \) and \( \sqrt[4]{4} \), we can express \( \sqrt[3]{3} \) in terms of base 2 as well.

### Step 3: Use logarithms to compare
We can use logarithms to compare the values:
1. For \( \sqrt{2} = 2^{1/2} \):
\[ \log(\sqrt{2}) = \frac{1}{2} \log(2) \]

2. For \( \sqrt[3]{3} = 3^{1/3} \):
\[ \log(\sqrt[3]{3}) = \frac{1}{3} \log(3) \]

3. For \( \sqrt[4]{4} = 2^{1/2} \):
\[ \log(\sqrt[4]{4}) = \frac{1}{4} \log(4) = \frac{1}{4} \cdot 2 \log(2) = \frac{1}{2} \log(2) \]

### Step 4: Compare the logarithmic values
Now we can compare:
- \( \log(\sqrt{2}) = \frac{1}{2} \log(2) \)
- \( \log(\sqrt[3]{3}) = \frac{1}{3} \log(3) \)
- \( \log(\sqrt[4]{4}) = \frac{1}{2} \log(2) \)

### Step 5: Calculate numerical values
We need to calculate the approximate values of these logarithms:
- \( \log(2) \approx 0.301 \)
- \( \log(3) \approx 0.477 \)

Calculating:
1. \( \log(\sqrt{2}) \approx \frac{1}{2} \cdot 0.301 \approx 0.1505 \)
2. \( \log(\sqrt[3]{3}) \approx \frac{1}{3} \cdot 0.477 \approx 0.159 \)
3. \( \log(\sqrt[4]{4}) \approx \frac{1}{2} \cdot 0.301 \approx 0.1505 \)

### Step 6: Conclusion
From the calculations:
- \( \log(\sqrt{2}) \approx 0.1505 \)
- \( \log(\sqrt[3]{3}) \approx 0.159 \)
- \( \log(\sqrt[4]{4}) \approx 0.1505 \)

Thus, \( \sqrt[3]{3} \) is the largest among the three.

### Final Answer:
The largest number among \( \sqrt{2} \), \( \sqrt[3]{3} \), and \( \sqrt[4]{4} \) is \( \sqrt[3]{3} \).

---