Question
Algebra 2A Semester Exam
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Question
What index should the root form of the expression 1258⋅12−12
have?(1 point)
Answers
GPT-4o mini
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**.
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**.