Radicals & Rational Exponents Quick Check

Question

3 of 53 of 5 Items
Question
Which of the following is an equivalent expression to 81−12
?(1 point)
Responses

−81−−√
negative Start Root 81 End Root

−181√
negative Start Fraction 1 over Start Root 81 End Root End Fraction

181√
Start Fraction 1 over Start Root 81 End Root End Fraction

81−−√

Answer 1

To simplify the expression \( 81^{-1/2} \), we can use the property of negative exponents and the property of rational exponents.

1. The expression \( 81^{-1/2} \) can first be rewritten using the negative exponent rule: \[ 81^{-1/2} = \frac{1}{81^{1/2}} \]

2. Now, we know that \( 81^{1/2} \) is the square root of 81: \[ 81^{1/2} = \sqrt{81} \] Since \( \sqrt{81} = 9 \), we can substitute that in: \[ 81^{-1/2} = \frac{1}{\sqrt{81}} = \frac{1}{9} \]

Now let's compare this with the given answers:

1. −81−−√: This does not match our result.
2. −181√: This is \( -\frac{1}{\sqrt{81}} \), which does not match since we have a positive \( \frac{1}{9} \).
3. 181√: This is \( \frac{1}{\sqrt{81}} \), which simplifies to \( \frac{1}{9} \). This matches our result.
4. 81−−√: This just gives us \( \sqrt{81} = 9 \), which does not match.

Thus, the equivalent expression to \( 81^{-1/2} \) is: \[ \frac{1}{\sqrt{81}} \quad \text{which is the third response: } \text{181√} \]