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

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

Answer 1

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

Calculating the exponents:

\[ m + n = -5 + (-11) = -16 \]

So, we have:

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

Now, to express this with positive exponents, we can apply the property \( a^{-n} = \frac{1}{a^n} \). Therefore:

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

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

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

None of the options directly state \(\frac{1}{13^{16}}\), but if we were to assume the options are not listed correctly or if we need to find a valid option, we can check the options provided:

1. \(\frac{1}{13^6}\)
2. \(\frac{1}{26^6}\)
3. \(\frac{1}{13^{16}}\) (not listed)
4. \(\frac{1}{26^{16}}\)

The answer is \(\frac{1}{13^{16}}\), which corresponds to other calculations as shown, but if strictly based on the options you mentioned, none match. The option to choose theoretically would be:

None of the options listed correctly reflect the simplification of \( 13^{-5} \cdot 13^{-11} \), which should be \(\frac{1}{13^{16}}\). If there was a typo or missing option for \(\frac{1}{13^{16}}\), that would be the correct equivalent expression with only positive exponents.

Answer 2

please pick one

Answer 3

Given the options provided and the requirement to choose one, I will select the closest answer based on the simplification we performed.

The closest equivalent expression in the options you provided is:

1/26^16

However, it is important to reiterate that the correct positive exponent form of the expression \( 13^{-5} \cdot 13^{-11} = \frac{1}{13^{16}} \) is not among the given options. If \(\frac{1}{13^{16}}\) was an option, that would have been the correct choice.

Since that isn't listed, I must choose:

1/26^16 as the answer from the provided options.