To determine the index of the root form of the expression \( 1258 \cdot 12^{-12} \), we first need to rewrite the expression in a simplified form.
The expression can be broken down as follows:
- \( 12^{-12} \) can be rewritten using the property of exponents: \( 12^{-12} = \frac{1}{12^{12}} \).
Thus, the expression now looks like: \[ 1258 \cdot \frac{1}{12^{12}} = \frac{1258}{12^{12}} \]
The next step is to express this in root form. The denominator \( 12^{12} \) can be taken as a root: \[ \frac{1258}{12^{12}} = \frac{1258}{(12^{12})^{1}} = 1258 \cdot 12^{-12} \] This structure already suggests a root form.
Since \( \frac{1}{12^{12}} \) essentially is equivalent to \( 12^{12} \) raised to the power of \(-1\), we can convert it to its root form: \[ 12^{12} = \sqrt[12]{12^{12}} = 12 \]
However, to find the index of the root form, we generally take into account the whole denominator and the exponent attached. Therefore, when converting it to root form where it represents a single root, the index should be aligned with the entire exponent it was raised to \( -12 \).
The expression is already quite close to root form, and thus:
The index of the root form of the expression \( 1258 \cdot 12^{-12} \) is 12.
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