Question
Radicals & Rational Exponents Quick Check
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−−√
Answers
GPT-4o mini
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:
\[
\frac{1}{\sqrt{81}} \quad \text{which is the third response: } \text{181√}
\]
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:
\[
\frac{1}{\sqrt{81}} \quad \text{which is the third response: } \text{181√}
\]