Which of the following is an equivalent expression to 59⋅5−13

Question

with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses

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

154
Start Fraction 1 over 5 superscript 4 baseline end fraction

1522
Start Fraction 1 over 5 superscript 22 baseline end fraction

54

Answer 1

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

First, we can express \( 5^9 \) as \( \frac{5^9}{5^{13}} \) using the property that \( a^m \cdot a^n = a^{m+n} \):

\[ 59 \cdot 5^{-13} = \frac{5^9}{1} \cdot \frac{1}{5^{13}} = \frac{5^9}{5^{13}} \]

Now, we can use the property \( \frac{a^m}{a^n} = a^{m-n} \):

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

Since we want an equivalent expression using only positive exponents, we can rewrite \( 5^{-4} \) as:

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

Now let's find which of the provided responses matches \( \frac{1}{5^4} \):

1. \( 595 - 13 \)
2. \( \frac{5^9}{5^{-13}} \)
3. \( \frac{1}{5^4} \)
4. \( \frac{1}{5^{22}} \)
5. \( 5^4 \)

The correct equivalent expression is:

\(\frac{1}{5^4}\)

Thus, the answer is:

\(\frac{1}{5^4}\)