To simplify the expression \( 5^9 \cdot 5^{-615} \) and find an equivalent expression with only positive exponents, we can use the properties of exponents.
According to the properties of exponents, when you multiply bases that are the same, you can add the exponents:
\[ 5^9 \cdot 5^{-615} = 5^{9 + (-615)} = 5^{9 - 615} = 5^{-606} \]
Now, we want to express \( 5^{-606} \) with positive exponents. The rule for converting a negative exponent to a positive one is:
\[ a^{-n} = \frac{1}{a^n} \]
Thus,
\[ 5^{-606} = \frac{1}{5^{606}} \]
Now let's check the options given:
- \( \frac{1}{5^4} \)
- \( \frac{5^9}{5^{-13}} \)
- \( 5^4 \)
- \( \frac{1}{5^{22}} \)
None of these options directly match \( \frac{1}{5^{606}} \).
Therefore, none of the provided options is equivalent to \( 5^9 \cdot 5^{-615} \) expressed with only positive exponents.
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