Which value among `root4(7)`, `root3(11)` and `root12(1257)` is the largest ?

Options

A. `root3(11)`
B. `root4(7)`
C. `root12(1257)`
D. All are equal

Answer

To determine which value among \( \sqrt[4]{7} \), \( \sqrt[3]{11} \), and \( \sqrt[12]{1257} \) is the largest, we can express each root in terms of exponents:

1. **Convert roots to exponents**:
- \( \sqrt[4]{7} = 7^{1/4} \)
- \( \sqrt[3]{11} = 11^{1/3} \)
- \( \sqrt[12]{1257} = 1257^{1/12} \)

2. **Find a common exponent**:
To compare these values, we can find a common denominator for the exponents. The least common multiple (LCM) of the denominators 4, 3, and 12 is 12.

3. **Rewrite each term with the common exponent**:
- For \( 7^{1/4} \):
\[ 7^{1/4} = 7^{3/12} \]
- For \( 11^{1/3} \):
\[ 11^{1/3} = 11^{4/12} \]
- For \( 1257^{1/12} \):
\[ 1257^{1/12} = 1257^{1/12} \]

4. **Compare the values**:
Now we have:
- \( 7^{3/12} \)
- \( 11^{4/12} \)
- \( 1257^{1/12} \)

To compare these, we can raise each base to the power of 12 to eliminate the fractional exponent:
- \( 7^3 = 343 \)
- \( 11^4 = 14641 \)
- \( 1257^1 = 1257 \)

5. **Determine the largest value**:
Now we compare:
- \( 343 \) (from \( 7^{3/12} \))
- \( 14641 \) (from \( 11^{4/12} \))
- \( 1257 \) (from \( 1257^{1/12} \))

Clearly, \( 14641 \) is the largest value.

6. **Conclusion**:
Therefore, \( \sqrt[3]{11} \) is the largest among the three values.