Question
Question
Which of the following is an equivalent expression to 13^−5 ⋅13^−11
with only positive exponents, generated by applying the properties of exponents?(1 point)
Responses
12616
Start Fraction 1 over 26 superscript 16 baseline end fraction
1136
Start Fraction 1 over 13 Superscript 6 baseline End Fraction
1266
Start Fraction 1 over 26 superscript 6 baseline end fraction
11316
Start Fraction 1 over 13 superscript 16 baseline end fraction
Answers
GPT-4o mini
To simplify the expression \( 13^{-5} \cdot 13^{-11} \), we can use the property of exponents that states \( a^m \cdot a^n = a^{m+n} \).
So we calculate:
\[
13^{-5} \cdot 13^{-11} = 13^{-5 + (-11)} = 13^{-16}
\]
We want to express this with positive exponents. To convert \( 13^{-16} \) to a positive exponent, we use the reciprocal property, which states that \( a^{-n} = \frac{1}{a^n} \). Therefore, we have:
\[
13^{-16} = \frac{1}{13^{16}}
\]
Now we need to compare this with the given responses:
1. \( \frac{1}{26^{16}} \)
2. \( \frac{1}{13^{6}} \)
3. \( \frac{1}{26^{6}} \)
4. \( \frac{1}{13^{16}} \)
The equivalent expression to \( 13^{-5} \cdot 13^{-11} \) with only positive exponents is:
\[
\frac{1}{13^{16}}
\]
Thus, the correct response is:
**\( \frac{1}{13^{16}} \)**.
So we calculate:
\[
13^{-5} \cdot 13^{-11} = 13^{-5 + (-11)} = 13^{-16}
\]
We want to express this with positive exponents. To convert \( 13^{-16} \) to a positive exponent, we use the reciprocal property, which states that \( a^{-n} = \frac{1}{a^n} \). Therefore, we have:
\[
13^{-16} = \frac{1}{13^{16}}
\]
Now we need to compare this with the given responses:
1. \( \frac{1}{26^{16}} \)
2. \( \frac{1}{13^{6}} \)
3. \( \frac{1}{26^{6}} \)
4. \( \frac{1}{13^{16}} \)
The equivalent expression to \( 13^{-5} \cdot 13^{-11} \) with only positive exponents is:
\[
\frac{1}{13^{16}}
\]
Thus, the correct response is:
**\( \frac{1}{13^{16}} \)**.