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

Question

Responses:

1 over 26^16

1 over 13^6

1 over 26^6

1 over 13^16

Answer 1

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

Applying this rule, we have:

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

To express this with only positive exponents, we use the property that \( a^{-n} = \frac{1}{a^n} \):

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

Now, we can compare this to the provided options. The equivalent expression to \( 13^{-5} \cdot 13^{-11} \) with only positive exponents is:

1 over 13^16.