Which of the following is an equivalent expression to 13^-5 x 13^-11 with only positive exponents, generated by applying the properties of exponents?

Question

1/26^6
1/13^6
1/26^16
1/13^16

Answer 1

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

So we add the exponents:

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

Now we want to express this with only positive exponents.

To convert \( 13^{-16} \) to a positive exponent, we use the property \( a^{-m} = \frac{1}{a^m} \):

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

Looking at the options provided:

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

The equivalent expression is:

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

Thus, the answer is:

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