Which of the following is an equivalent expression to 81−1/2 ?

Question

Responses

181√
the fraction with numerator 1 and denominator square root of 81 - not selected, this is the correct answer

−181√
negative the fraction with numerator 1 and denominator square root of 81 - no response given

81−−√
square root of 81 - no response given

−81−−√
and show how to solve

Answer 1

To simplify the expression \( 81^{-1/2} \), we can use the rules of exponents. The expression can be broken down as follows:

1. Understanding the exponent: The exponent of \(-1/2\) indicates two things:

   • The negative sign means we take the reciprocal.
   • The fraction means we are dealing with a square root.

2. Rewrite the expression: \[ 81^{-1/2} = \frac{1}{81^{1/2}} \]

3. Evaluate \( 81^{1/2} \): The square root of \( 81 \) is \( 9 \) because \( 9 \times 9 = 81 \). Therefore: \[ 81^{1/2} = 9 \]

4. Substitute back into the expression: Now substitute \( 9 \) back into our rewritten expression: \[ 81^{-1/2} = \frac{1}{9} \]

Thus, we can conclude that:

The equivalent expression to \( 81^{-1/2} \) is \( \frac{1}{\sqrt{81}} \), which evaluates to \( \frac{1}{9} \).

If we focus on the list of given responses, the equivalent expression stated as "the fraction with numerator 1 and denominator square root of 81" correctly represents the solution since:

\[ \sqrt{81} = 9 \quad \text{and} \quad \frac{1}{\sqrt{81}} = \frac{1}{9} \]

This confirms that the correct response is: the fraction with numerator 1 and denominator square root of 81.