Which one among `root3(6)`, `root2(5)` and `root6(12)` is the largest ?

Options

A. `root3(6)`
B. `root2(5)`
C. `root6(12)`
D. All are equal

Answer

To determine which among \( \sqrt[3]{6} \), \( \sqrt{5} \), and \( \sqrt[6]{12} \) is the largest, we can compare these values by expressing them with a common exponent.

### Step 1: Rewrite each expression with a common exponent
We will express each root with a common denominator for the exponents. The denominators we have are 3, 2, and 6. The least common multiple (LCM) of these numbers is 6.

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

- For \( \sqrt{5} \):
\[ \sqrt{5} = 5^{1/2} = 5^{3/6} \]

- For \( \sqrt[6]{12} \):
\[ \sqrt[6]{12} = 12^{1/6} \]

### Step 2: Compare the values raised to the power of 6
Now we can rewrite the expressions:
- \( 6^{2/6} \) becomes \( 6^2 = 36 \)
- \( 5^{3/6} \) becomes \( 5^3 = 125 \)
- \( 12^{1/6} \) becomes \( 12^1 = 12 \)

### Step 3: Evaluate and compare
Now we compare the values:
- \( 6^{2} = 36 \)
- \( 5^{3} = 125 \)
- \( 12^{1} = 12 \)

### Step 4: Identify the largest value
Among these calculated values:
- \( 36 \)
- \( 125 \)
- \( 12 \)

The largest value is \( 125 \).

### Conclusion
Thus, \( \sqrt{5} \) is the largest among \( \sqrt[3]{6} \), \( \sqrt{5} \), and \( \sqrt[6]{12} \).

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