Negative Integer Exponents Quick Check

Question

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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

1266
Start Fraction 1 over 26 superscript 6 baseline end fraction

12616
Start Fraction 1 over 26 superscript 16 baseline end fraction

11316
Start Fraction 1 over 13 superscript 16 baseline end fraction

1136

Answer 1

To simplify the expression \( 13^{-5} \cdot 13^{-11} \) using the properties of exponents, we can use the property that states \( a^m \cdot a^n = a^{m+n} \).

So, we have:

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

Now we want to express this with positive exponents. We can do this by utilizing the property for negative exponents, which states \( a^{-n} = \frac{1}{a^n} \). Thus,

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

Now let's analyze the answer choices:

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

The correct equivalent expression with only positive exponents is:

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

Thus, the right answer is:

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