Which of the following is the greatest?

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Accepted Answer

To compare $$ \sqrt{64} $$ and $$ \sqrt{81} $$:

1. **Express in exponential form**:  
   - $$ \sqrt{64} = 64^{\frac{1}{12}} = (2^6)^{\frac{1}{12}} = 2^{\frac{6}{12}} = 2^{\frac{1}{2}} = \sqrt{2} $$.  
   - $$ \sqrt{81} = 81^{\frac{1}{12}} = (3^4)^{\frac{1}{12}} = 3^{\frac{4}{12}} = 3^{\frac{1}{3}} = \sqrt{3} $$.

2. **Compare the values**:  
   From the calculations, $$ \sqrt{3} > \sqrt{2} $$, as the cube root of 3 is greater than the square root of 2.

Thus, **$$ \sqrt{81} $$** is the greatest.

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