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

11316
Start Fraction 1 over 13 superscript 16 baseline end fraction

1266
Start Fraction 1 over 26 superscript 6 baseline end fraction

1136
Start Fraction 1 over 13 Superscript 6 baseline End Fraction

12616

1 answer

To simplify the expression \( 13^{-5} \cdot 13^{-11} \), we can apply the properties of exponents. Specifically, when multiplying like bases, we can add the exponents:

\[ 13^{-5} \cdot 13^{-11} = 13^{-5 + (-11)} = 13^{-16} \]

Next, we want to express this in terms of positive exponents. Using the property that \( a^{-n} = \frac{1}{a^n} \), we can rewrite \( 13^{-16} \) as:

\[ 13^{-16} = \frac{1}{13^{16}} \]

Looking through the provided options, the equivalent expression with only positive exponents is:

\[ \frac{1}{13^{16}} \]

So the correct response is:

Start Fraction 1 over 13 superscript 16 baseline end fraction.