Which of the following is an equivalent expression to 13^−5 ⋅13^−11 with only positive exponents, generated by applying the properties of exponents?1/26^6

Question

1/26^16

1/13^16

1/1^36

Answer 1

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

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

Now we want to express this with only positive exponents. The rule for converting a negative exponent to a positive exponent is:

\[ a^{-n} = \frac{1}{a^n} \]

So, we have:

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

Thus, the equivalent expression with only positive exponents is:

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

Looking at the given options, the correct equivalent expression is:

1/13^16.