To simplify the expression \(\frac{12^{5/8}}{12^{-1/2}}\), we can use the properties of exponents. Specifically, when dividing two expressions with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
\[ \frac{12^{5/8}}{12^{-1/2}} = 12^{5/8 - (-1/2)} = 12^{5/8 + 1/2} \]
Now, we need to convert \(\frac{1}{2}\) to a fraction with a common denominator of 8:
\[ 1/2 = 4/8 \]
Thus, we can rewrite the expression as:
\[ 12^{5/8 + 4/8} = 12^{(5 + 4)/8} = 12^{9/8} \]
So, the simplified expression is \(12^{9/8}\).
The index of the root (or exponent) form is \(9/8\). Therefore, the index of the root form of the expression \( \frac{12^{5/8}}{12^{-1/2}} \) is:
\[ \boxed{\frac{9}{8}} \]
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