To generate an equivalent expression for [tex]\( 3^{\frac{-1}{2}} \)[/tex] using the Negative Exponent Rule, follow these steps:
1. Understand the Negative Exponent Rule: The rule states that [tex]\( a^{-b} = \frac{1}{a^b} \)[/tex]. This means that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.
2. Apply the Rule: For the given expression [tex]\( 3^{\frac{-1}{2}} \)[/tex]:
[tex]\[
3^{\frac{-1}{2}} = \frac{1}{3^{\frac{1}{2}}}
\][/tex]
3. Simplify the Expression: Recognize that [tex]\( 3^{\frac{1}{2}} \)[/tex] represents the square root of 3:
[tex]\[
3^{\frac{1}{2}} = \sqrt{3}
\][/tex]
4. Combine the Steps: Substitute back into the equation:
[tex]\[
3^{\frac{-1}{2}} = \frac{1}{\sqrt{3}}
\][/tex]
The equivalent expression using the Negative Exponent Rule is [tex]\( \frac{1}{\sqrt{3}} \)[/tex].
Evaluating this expression, we find that the numerical result is approximately [tex]\( 0.5773502691896258 \)[/tex].
