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)

Question

Responses

595−13
Start Fraction 5 superscript 9 baseline over 5 superscript negative 13 baseline End Fraction

54
5 superscript 4 baseline

1522
Start Fraction 1 over 5 superscript 22 baseline end fraction

154

Answer 1

To simplify the expression \(59 \cdot 5^{-13}\) using the properties of exponents, we can rewrite it as:

\[ 59 \cdot 5^{-13} = 5^{9} \cdot 5^{-13} \]

Using the property of exponents that states \(a^{m} \cdot a^{n} = a^{m+n}\), we combine the exponents:

\[ 5^{9} \cdot 5^{-13} = 5^{9 - 13} = 5^{-4} \]

Now, to express \(5^{-4}\) with only positive exponents, we use the property that \(a^{-n} = \frac{1}{a^{n}}\):

\[ 5^{-4} = \frac{1}{5^{4}} \]

Now, let's look at your options:

1. \(59 \cdot 5^{-13}\)
2. \(\frac{5^{9}}{5^{-13}}\)
3. \(5^{4}\)
4. \(\frac{1}{5^{22}}\)
5. \(5^{4}\)

From our simplification, we found that the expression simplifies to \(\frac{1}{5^{4}}\). However, looking at the provided options, we see the form \(\frac{1}{5^{22}}\) and \(5^{4}\) are not correct representations of our expression.

Since none of the options directly state \(5^{-4}\) or \(\frac{1}{5^{4}}\), if you meant to ask which might align the closest, then the correct form in a simplified and positive context \(\frac{1}{5^{4}}\) is not explicitly stated but \(5^4\) alone is incorrect for representing the negative exponent.

Please confirm the question for accurate response selection. If focusing on derived forms only, stating 'none' could also apply.