Which of the following is an equivalent expression to 59⋅5−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)

To simplify the expression \( 59 \cdot 5^{-13} \) to an equivalent expression with only positive exponents, we can apply the Properties of Integer Exponents.

1. Start with the expression:
\[
59 \cdot 5^{-13}
\]

2. Recall that \( 5^{-13} = \frac{1}{5^{13}} \). So we can rewrite the expression:
\[
59 \cdot 5^{-13} = 59 \cdot \frac{1}{5^{13}} = \frac{59}{5^{13}}
\]
This expression still has \( 5^{-13} \), so let's keep going.

3. Since no option can be simplified to just \( 59 \) without involving the denominator \( 5^{13} \) or some manipulation, we look for any of the provided options that match this format.

Now, we analyze the options provided:

1. \( 54 \) \(\rightarrow\) This does not match.
2. \( 1/5^{22} \) \(\rightarrow\) This does not match either.
3. \( 5^{9}/5^{-13} \) \(\rightarrow\) This can be simplified. Using the properties of exponents, we get:
\[
\frac{5^{9}}{5^{-13}} = 5^{9 - (-13)} = 5^{9 + 13} = 5^{22}
\]
This involves getting the value 5's powers which doesn't return to the original expression.
4. \( 1/54 \) \(\rightarrow\) This does not match.

None of the provided options directly match, but focusing on the closest, we find the first option transforms correctly into \( \frac{59}{5^{13}} \).

Given the information, the correct transformation you were looking for is indeed among those options, and interpreting them confirms:

- \( \frac{59}{5^{13}} \) is not represented but could relate to \( \frac{5^9}{5^{-13}} \), which converts to \( 5^{22} \) as mentioned.

However, since transformations did not yield a match correctly among the integers provided options, the one that stands out related to exponent properties is:
\[
5^{9}/5^{-13} \; \text{ resolves in the context of properties.}
\]
Thus indicating the relevance of these operations even if not seeing \( \frac{59}{\text{}} \) or full match.

**The option that relates the best in transformation due to properties and rework ensuing is:**

5. \( \frac{5^9}{5^{-13}} \) => **valid simplification based on operations.**