Which of the following is an equivalent expression to 13 −5 ⋅13 −11

Question

with only positive exponents, generated by applying the properties of exponents?(1 point)
Responses

1/26^6
Start Fraction 1 over 26 superscript 6 baseline end fraction

1/26^16
Start Fraction 1 over 26 superscript 16 baseline end fraction

1/13^16
Start Fraction 1 over 13 superscript 16 baseline end fraction

1/13^6

Answer 1

To simplify the expression \( 13^{-5} \cdot 13^{-11} \) and express it with positive exponents, we can use the properties of exponents. Specifically, we apply the rule that states \( a^m \cdot a^n = a^{m+n} \).

1. First, we combine the exponents: \[ 13^{-5} \cdot 13^{-11} = 13^{-5 + (-11)} = 13^{-16}. \]

2. Now, to express \( 13^{-16} \) with a positive exponent, we use the property that states \( a^{-n} = \frac{1}{a^n} \): \[ 13^{-16} = \frac{1}{13^{16}}. \]

Now let's look at the provided options:

1. \( \frac{1}{26^6} \)
2. \( \frac{1}{26^{16}} \)
3. \( \frac{1}{13^{16}} \)
4. \( \frac{1}{13^6} \)

Among the options, \( \frac{1}{13^{16}} \) matches our simplified expression. Thus, the answer is:

\( \frac{1}{13^{16}} \).