To simplify the expression \( 12^{5/8} \cdot 12^{-1/2} \), you can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
So, add the exponents:
\[ \frac{5}{8} + \left(-\frac{1}{2}\right) \]
To add these fractions, you need a common denominator. The common denominator between 8 and 2 is 8. Rewrite \(-\frac{1}{2}\) as \(-\frac{4}{8}\):
\[ \frac{5}{8} - \frac{4}{8} = \frac{1}{8} \]
Now, the expression simplifies to:
\[ 12^{\frac{1}{8}} \]
Thus, the root form of the expression \( 12^{5/8} \cdot 12^{-1/2} \) is:
\[ 12^{\frac{1}{8}} = \sqrt[8]{12} \]
The index of the root form is \( 8 \).
Answer: 8
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