Which value among the `root3(8)`, `root2(4)`, `root6(64)` is greatest ?

Options

A. `root3(8)`
B. `root2(4)`
C. `root6(64)`
D. All are equal

Answer

To determine which value among \( \sqrt[3]{8} \), \( \sqrt{4} \), and \( \sqrt[6]{64} \) is greatest, we can express each root as a power of its base and then compare them.

### Step-by-Step Solution:

1. **Convert each root to exponent form**:
- \( \sqrt[3]{8} = 8^{\frac{1}{3}} \)
- \( \sqrt{4} = 4^{\frac{1}{2}} \)
- \( \sqrt[6]{64} = 64^{\frac{1}{6}} \)

2. **Rewrite the bases as powers of 2**:
- \( 8 = 2^3 \), so \( 8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2 \)
- \( 4 = 2^2 \), so \( 4^{\frac{1}{2}} = (2^2)^{\frac{1}{2}} = 2^{2 \cdot \frac{1}{2}} = 2^1 = 2 \)
- \( 64 = 2^6 \), so \( 64^{\frac{1}{6}} = (2^6)^{\frac{1}{6}} = 2^{6 \cdot \frac{1}{6}} = 2^1 = 2 \)

3. **Compare the values**:
- From our calculations, we have:
- \( \sqrt[3]{8} = 2 \)
- \( \sqrt{4} = 2 \)
- \( \sqrt[6]{64} = 2 \)

4. **Conclusion**:
- All three values are equal to 2.

### Final Answer:
Thus, the greatest value among \( \sqrt[3]{8} \), \( \sqrt{4} \), and \( \sqrt[6]{64} \) is that they are all equal.

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