Which of the following is the largest number among `sqrt(2)`, `root3(3)`, `sqrt(4)` and `root3(5)` ?

Options

A. `sqrt(2)`
B. `root3(3)`
C. `sqrt(4)`
D. `root3(5)`

Answer

To determine which of the following numbers is the largest among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt{4} \), and \( \sqrt[3]{5} \), we can follow these steps:

### Step 1: Rewrite the numbers in exponent form
We can express each of the numbers using exponents:
- \( \sqrt{2} = 2^{1/2} \)
- \( \sqrt[3]{3} = 3^{1/3} \)
- \( \sqrt{4} = 4^{1/2} = (2^2)^{1/2} = 2^1 = 2 \)
- \( \sqrt[3]{5} = 5^{1/3} \)

### Step 2: Find a common exponent
To compare these numbers, we can find a common exponent. The least common multiple (LCM) of the denominators of the exponents (2 and 3) is 6. Therefore, we will convert each number to the power of 6:
- \( \sqrt{2} = 2^{1/2} = 2^{3/6} \)
- \( \sqrt[3]{3} = 3^{1/3} = 3^{2/6} \)
- \( \sqrt{4} = 2^1 = 2^{6/6} \)
- \( \sqrt[3]{5} = 5^{1/3} = 5^{2/6} \)

### Step 3: Calculate the values
Now we can calculate the equivalent values:
- \( 2^{3/6} = 2^{0.5} \)
- \( 3^{2/6} = 3^{0.3333} \)
- \( 2^{6/6} = 2^{1} = 2 \)
- \( 5^{2/6} = 5^{0.3333} \)

### Step 4: Evaluate the powers
Now we can evaluate these powers:
- \( 2^{3/6} = \sqrt{2} \approx 1.414 \)
- \( 3^{2/6} = \sqrt[3]{3} \approx 1.442 \)
- \( 2^{6/6} = 2 \)
- \( 5^{2/6} = \sqrt[3]{5} \approx 1.710 \)

### Step 5: Compare the values
Now we compare the approximate values:
- \( \sqrt{2} \approx 1.414 \)
- \( \sqrt[3]{3} \approx 1.442 \)
- \( \sqrt{4} = 2 \)
- \( \sqrt[3]{5} \approx 1.710 \)

### Conclusion
From the calculations, we can see that \( \sqrt{4} = 2 \) is the largest number among \( \sqrt{2} \), \( \sqrt[3]{3} \), \( \sqrt{4} \), and \( \sqrt[3]{5} \).

### Final Answer
The largest number is \( \sqrt{4} \).

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