Which of the following is the biggest ?
`root3(4)`, `root4(6)`, `root6(15)`, and `root12(245)`,

Options

A. `root3(4)`
B. `root4(6)`
C. `root6(15)`
D. `root12(245)`

Answer

To determine which of the following expressions is the biggest: \( \sqrt[3]{4} \), \( \sqrt[4]{6} \), \( \sqrt[6]{15} \), and \( \sqrt[12]{245} \), we can follow these steps:

### Step 1: Identify the roots and their powers
The expressions can be rewritten with their respective powers:
- \( \sqrt[3]{4} = 4^{1/3} \)
- \( \sqrt[4]{6} = 6^{1/4} \)
- \( \sqrt[6]{15} = 15^{1/6} \)
- \( \sqrt[12]{245} = 245^{1/12} \)

### Step 2: Find the least common multiple (LCM) of the denominators
The denominators of the powers are 3, 4, 6, and 12. The LCM of these numbers is 12.

### Step 3: Rewrite each expression with the LCM as the denominator
Now, we will express each term with the LCM (12) as the denominator:
- \( 4^{1/3} = 4^{4/12} \)
- \( 6^{1/4} = 6^{3/12} \)
- \( 15^{1/6} = 15^{2/12} \)
- \( 245^{1/12} = 245^{1/12} \)

### Step 4: Calculate the equivalent powers
Now we will calculate the bases raised to the new powers:
- \( 4^{4/12} = 4^4 = 256 \)
- \( 6^{3/12} = 6^3 = 216 \)
- \( 15^{2/12} = 15^2 = 225 \)
- \( 245^{1/12} = 245^1 = 245 \)

### Step 5: Compare the results
Now we compare the results:
- \( 256 \) (from \( \sqrt[3]{4} \))
- \( 216 \) (from \( \sqrt[4]{6} \))
- \( 225 \) (from \( \sqrt[6]{15} \))
- \( 245 \) (from \( \sqrt[12]{245} \))

The largest value is \( 256 \).

### Conclusion
Thus, the biggest expression among \( \sqrt[3]{4} \), \( \sqrt[4]{6} \), \( \sqrt[6]{15} \), and \( \sqrt[12]{245} \) is \( \sqrt[3]{4} \).

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